Cleveland State University Cleveland State University

EngagedScholarship@CSU EngagedScholarship@CSU

ETD Archive

2015

Time and Temperature Dependent Surface Tension Time and Temperature Dependent Surface Tension

Measurements of Responsive Protein-Based Polymer Surfactant Measurements of Responsive Protein-Based Polymer Surfactant

Solutions Solutions

Hakan Celik Cleveland State University

Follow this and additional works at: https://engagedscholarship.csuohio.edu/etdarchive

Part of the Biomedical Engineering and Bioengineering Commons

How does access to this work benefit you? Let us know! How does access to this work benefit you? Let us know!

Recommended Citation Recommended Citation Celik, Hakan, "Time and Temperature Dependent Surface Tension Measurements of Responsive Protein-Based Polymer Surfactant Solutions" (2015). ETD Archive. 470. https://engagedscholarship.csuohio.edu/etdarchive/470

This Thesis is brought to you for free and open access by EngagedScholarship@CSU. It has been accepted for inclusion in ETD Archive by an authorized administrator of EngagedScholarship@CSU. For more information, please contact library.es@csuohio.edu.

Time and Temperature Dependent Surface Tension Measurements of

Responsive Protein-based Polymer Surfactant Solutions

HAKAN CELIK

Bachelor of Science in Physics

Dicle University

July 2005

submitted in partial fulfillment of requirements for the degree

MASTER OF SCIENCE IN BIOMEDICAL ENGINEERING

at the

CLEVELAND STATE UNIVERSITY

August 2015

We hereby approve this thesis for

Hakan Celik

Candidate for the Master of Science in Biomedical Engineering degree for the

Department of Chemical and Biomedical Engineering

and the CLEVELAND STATE UNIVERSITY

College of Graduate Studies

_________________________________________________________________

Nolan B. Holland, PhD

Thesis Chairperson

Department of Chemical and Biomedical Engineering

____________________

Date

_________________________________________________________________

Sridhar Ungarala, PhD

Department of Chemical and Biomedical Engineering

____________________

Date

_________________________________________________________________

Dhananjai B. Shah, PhD

Department of Chemical and Biomedical Engineering

____________________

Date

Student’s Date of Defense: 5 August 2015

ACKNOWLEGMENTS

I would like to convey my deep appreciation to Dr. Nolan B. Holland for the

opportunities he afforded me in the laboratory. His support and guidance inside the

laboratory as well as other facets of my education have been invaluable. I am incredibly

thankful to Dr. Metin Aytekin for advice, support and encouragement.

I would like to extend my appreciation to my committee members Dr. Dhananjai

B. Shah and Dr. Ungarala for the encouragement and advice throughout my studies. The

staff and faculty of the Biomedical Engineering Department at Cleveland State

University have provided an encouraging and comfortable environment in which to

study. I am grateful for all that the department has done to facilitate learning and

cooperation. My colleagues at the laboratory provided support for which I extend my

gratitude.

Ms. Becky Laird and Ms. Darlene Montgomery have provided invaluable support

throughout my studies, many thanks for everything. Thank you to Dr. Jim Cole, Eric

Helm, Jack Gavin, and Ryan Martin for all the advice and encouragement in my work. I

extend my gratitude to Dr. Orhan Talu and Dr. Joanne Belovich for all the

encouragement and advice provided.

Finally, I would like to express my appreciation of Dr. Veysel Celik for all of the

academic and moral support. I would like to thank Holly Korovich and Adam Maraschky

for being supportive friends and providing language support. My I would like to extend

my gratitude for all the support and motivation from my family.

iv

TIME AND TEMPERATURE DEPENDENT SURFACE TENSION

MEASUREMENTS OF RESPONSIVE PROTEIN-BASED POLYMER

SURFACTANT SOLUTIONS

HAKAN CELIK

ABSTRACT

A three-armed star elastin-like polypeptide (ELP-foldon) has thermoreversible

character which exhibits phase separation above a transition temperature (Tt) in

physiologic salt concentrations. At lower salt concentration, the ELP-foldon behaves like

a thermoresponsive surfactant, exhibiting micelle formation above its Tt. The purpose of

this study is characterize the surfactant behavior of the ELP-foldon at air-liquid interface

by measuring the surface tension. The surface tension is measured as a function of time

for different ELP concentrations from 10 nM to 50 μM and over range of temperatures

from 25 to 35 ℃ using the axisymmetric drop shape analysis (ADSA). The ADSA is a

method which is based on the analysis of the shape and size of drop or bubble profiles to

measure surface tension.

It has been determined that the surface tension is not different between conditions

where there are no micelles and where micelle form. Therefore, a critical micelle

concentration (c.m.c.) measurement by surface tension is not meaningful. The surface

tension exhibits a time-dependent reduction which can be fit with the Hua-Rosen

equation. The meso-equilibrium surface pressure is ~23 mN/m and does not vary with the

bulk concentration or the temperature. The time to reach the meso-equilibrium does vary

with the bulk concentration. These times scale with concentration by a power of -1.2 and

-1.3, suggesting that the process is not fully diffusion limited.

v

TABLE OF CONTENTS

ABSTRACT ....................................................................................................................... iv

LIST OF TABLES ............................................................................................................ vii

LIST OF FIGURES ......................................................................................................... viii

CHAPTER I ........................................................................................................................ 1

INTRODUCTION AND BACKGROUND ....................................................................... 1

1.1 Introduction ............................................................................................................... 1

1.2 Protein Adsorption .................................................................................................... 2

1.3 Polymer Adsorption .................................................................................................. 7

1.4 Surfactants (surface active agent) ............................................................................. 8

1.4.1 Anionic surfactants ............................................................................................. 9

1.4.2 Cationic surfactant .............................................................................................. 9

1.4.3 Nonionic surfactant ........................................................................................... 10

1.4.4 Zwitterion surfactant......................................................................................... 10

1.5 Micelle formation .................................................................................................... 10

1.5.1 Thermodynamic of Micelle Formation ............................................................. 11

1.5.2 Parameters impacting c.m.c .............................................................................. 13

1.5.2.1 Impact of Temperature and Pressure ......................................................... 13

1.5.2.2 Impact of Added Salt ................................................................................. 13

1.5.2.3 Impact of Head Group and Chain Length ................................................. 14

1.5.2.4 Impact of Organic Molecules..................................................................... 15

vi

1.6 ELP –Foldon (MGH(GVGVPGEGVP)(GVGVP)41GWP-Foldon) ........................ 15

1.7 Thermodynamics of ELP-foldon ............................................................................. 16

CHAPTER II ..................................................................................................................... 21

MATERIALS AND METHOD ........................................................................................ 21

2.1 Expression and Purification of ELP-Foldon ........................................................... 21

2.2 Surface Tension Apparatus ..................................................................................... 23

2.3 Axisymmetric Drop Shape Analysis (ADSA) ........................................................ 24

2.3.1 Image Capture ................................................................................................... 25

2.3.2 Image Analysis ................................................................................................. 26

2.3.2.1 Edge Detection ........................................................................................... 26

2.3.3 Drop Shape Calculation .................................................................................... 32

2.3.4 Optimization ..................................................................................................... 36

CHAPTER III ................................................................................................................... 42

RESULTS AND DISCUSSION ....................................................................................... 42

3.1 ELP-Foldon’s Diffusion .......................................................................................... 56

CHAPTER IV ................................................................................................................... 58

CONCLUSIONS............................................................................................................... 58

BIBLIOGRAPHY ............................................................................................................. 60

APPENDIX ....................................................................................................................... 65

vii

LIST OF TABLES

Table 1.1 Sum of the free energy change of the system. .................................................. 20

Table 2.1 The sample image’s result ................................................................................ 39

Table 3.1 The solution and the solvent surface tension values. ........................................ 50

Table 3.2 Fit Parameters for Linear and Trimer Constructs ............................................. 56

viii

LIST OF FIGURES

Figure 1 Dilute solution of typical proteins’ surface tension change ................................. 4

Figure 2 Surface pressure is a function of the fractional coverage using a Langmuir

adsorption model .................................................................................................. 7

Figure 3 Surfactant is with a triple structure ....................................................................... 8

Figure 4 The polymer’s arms aggregation with respect totemperature ............................ 16

Figure 5 The surfactant entropy change ............................................................................ 18

Figure 6 General procedure of Axisymmetric Drop Shape Analysis (ADSA). ................ 25

Figure 7 Schematic diagram of the experimental setup of ADSA.................................... 25

Figure 8 Charter to obtained final drop profile ................................................................. 26

Figure 9 Drop image for steps of image analysis process ................................................ 27

Figure 10 The kernels’ horizontal (Gx) and vertical (Gy) derivative approximations ...... 28

Figure 11 Drop profile’s cut-off point .............................................................................. 30

Figure 12 Midpoint of drop profile and midpoint of each curve point. ............................ 31

Figure 13 The distribution of the parameters geometrically on the drop ......................... 32

Figure 14 Drop profile analysis is used for the fitting process ......................................... 36

Figure 15 Theoretical and experimental curve are overlapped ......................................... 38

Figure 16 The minimum value of x against the value of b and c are described ............... 39

Figure 17 Comparison of literature values of surface tension of water and experimental

values of water is function of the temperature. ................................................ 41

Figure 18 Measured surface tension is a function of temperature for 10, 31.6, 0.1 μM ... 43

Figure 19 Surface tension is a function of time and temperature for a 0.2 μM ................ 45

ix

Figure 20 Surface tension is a function of time and temperature for a 0.316 μM ............ 46

Figure 24 Surface tension is a function of time and temperature for a 0.1 μM. ............... 47

Figure 22 Surface tension and rate of absorption (UV) is a function of time and

temperature for a 50 μM .................................................................................. 49

Figure 23 The surface pressure is a function of the temperature at 50 μM ...................... 50

Figure 24 The surface tension is a function of temperature for all concentrations and a

function of concentration at different temperatures ......................................... 51

Figure 25 Experimental data is fitted by Hua-Rosen equation to create theoretical curve

for 0.2, 0.316, 1, 50 μM. .................................................................................. 53

Figure 26 The half time (t*) is function of concentration for 0.2, 0.316, 1, 50 μM ......... 54

Figure 27 The polymers’ movement to air-aqueous surface ............................................ 56

1

CHAPTER I

INTRODUCTION AND BACKGROUND

1.1 Introduction

Proteins are large organic molecules formed in cells by binding amino acids to

each other to form chains. Proteins are important for living organisms because almost

every functional property of living organisms is performed by proteins. The structures of

proteins are defined by genes which have an important role in the protein synthesis. Cells

use the information in genes to produce all of the different protein structures for living

organisms. The genes can be modified to synthesize protein-based polymers in living

organisms.

Elastin-like polypeptides (ELP) are one such class of protein-based polymers

which can be synthesized using molecular biology techniques to generate recombinant

DNA (rDNA) molecules [1]. These protein-based polymers have been synthesized with

the desired structure and precision control [1, 2]. The ELP sequence is based on a

sequence in the elastin protein which can be found in blood, vessels, lungs, and skin [1].

ELPs are used for many applications, including tissue engineering and pharmaceutical

and biomedical sciences since ELPs are generally non-toxic, biocompatible,

biodegradable, and have good pharmacokinetics properties [1, 3].

2

Since ELP is a thermally responsive polymer which exhibits phase separation

above a transition temperature (Tt), ELP conformation is changed by the temperature.

Conformations of polymers are significantly affected by chemical and physical

characteristics of the polymers.

Molecular weight and length of polymer chain are significantly important for

physical properties of the polymer. To give an example, long polymer chain provides

excellent wear resistance and impacts toughness in ultra-high-molecular-weight

polyethylene (UHMWPE) [4]. Also, characterization of small polymers characterizations

is easier than large polymers [5]. Furthermore, molecular size of polymers affects surface

tension and surface tension change process. Surface tension can be defined as the energy

which is needed to increase the surface area of a liquid. Surface tension is affected by

surfactant, and some polymers are used as surfactants for liquid solutions.

1.2 Protein Adsorption

Polymers have long chain structures that are formed by connecting monomers via

chemical bonds [5]. Amino acids are the monomers that form the polymer structure of

proteins and polypeptides [5]. An important difference between proteins and

polypeptides is that the proteins have a defined conformational folded structure which

can be denatured through conformational changes; however, a polypeptide typically does

not have a single structure, but rather assumes a random coil conformation. Therefore,

adsorption kinetics of proteins and polypeptides are different.

Prior studies on protein solutions show protein film formation at the fluid

interface by the adsorption of the proteins to the surface from the bulk solution [6]. The

3

formation of films causes the reduction of the surface tension, increasing surface pressure

(𝛱) defined by

𝛱 = 𝛾0 − 𝛾

where, 𝛾0 is the solvent surface tension, and 𝛾 is the solution surface tension. Thereby,

the increase in the surface pressure is the result of a decrease of the surface tension during

protein adsorption. Protein adsorption occurs in the three steeps (diffusion, adsorption,

and rearrangement). Firstly, the proteins diffuse to the interface. After the proteins reach

to the interface, the state change of the proteins cause energy reduction in the system, and

in this case, the proteins want to further minimize free energy by conformational

rearrangement. The adsorption processes affect the time required for the system to reach

reduced interfacial tensions.

The reduction of surface tension as a function of time during protein adsorption

has been shown to exhibit an s-type curve for many proteins (Figure 1) [6]. The behavior

is only observed in dilute protein solution. In higher protein concentrations, it is not

observed, because, the proteins which are in the region close to surface reach to the

surface quickly to generate high surface coverage [7].

4

One protein they studied was ovalbumin a globular protein which has a molecular

weight of 42 kDa, one disulfide bond, and an isoelectric point of 4.6 [6]. Denaturation of

the ovalbumin by urea resulted in an increased time for the proteins to reach the interface

with respect to the urea-free system, since both kinetics of the protein adsorption was

reduced by denaturation. It has been suggested that the conformation resulted in higher

flexibility of molecules in the interface that had increased rearrangement time [6].

Higher ionic strength of the solvent can affect the regimes by reducing Debye

length of charged protein side groups, reducing the repulsion [6, 8] between the proteins.

The protein diffusion velocity is increased, resulting in faster interfacial saturation. Since

𝛽 − 𝑐𝑎𝑠𝑒𝑖𝑛 is a protein without disulfide bonds, it has a more disordered structure when

denatured, and this gives it flexible properties [6]. Prior studies show that 𝛽 − 𝑐𝑎𝑠𝑒𝑖𝑛

reach the second and third regime faster than ovalbumin since globular protein interfacial

Figure 1 Dilute solution of typical proteins’ surface tension change as depending

on time is illustrated. While the proteins adsorbed to the surface, the system shows

three different regimes. Modified from Beverung et al. [6].

5

unfolding and rearrangement in the solution is slow and the processes take extra time for

induction regime [6].

Regime I is an induction regime for the interfacial tension change and it is usually

equal to the pure solution interfacial tension at low protein concentrations. A theoretical

model for dynamics of interfacial tension (diffusion controlled adsorption kinetics) can

was developed by Ward and Tordai [9]. In the model, the effects of diffusion on

interfacial concentration 𝛤(𝑡) depends on bulk protein concentration (𝐶𝑏), and diffusion

coefficient (𝐷), the relation is [9]

𝛤(𝑡) = 2𝐶𝑏√𝐷𝑡

𝜋

where, back diffusion is negligible [6]. Regime I formation can also be explained by

Langmuir adsorption isotherm with Gibbs equation [6]

𝛱(𝑡) = −𝑘𝐵𝑇𝛤𝑚𝑎𝑥𝑙𝑛 (1 −𝛤(𝑡)

𝛤𝑚𝑎𝑥)

where 𝛱 is the surface pressure, 𝑘𝐵 is Boltzmann’s constant, 𝑇 is the temperature, 𝛤(𝑡) is

the molecular surface concentration at the time t, and 𝛤𝑚𝑎𝑥 the molecular surface

concentration at the maximum coverage [6]. To be seen in the equation, the surface

pressure is increased by the fractional coverage 𝛩 = 𝛤

𝛤𝑚𝑎𝑥 [6]. However, surface pressure

also depends on molecular size of the surfactants. In the equation, the area covered by an

adsorbing molecule in the interface is described as 1

𝛤𝑚𝑎𝑥 , and molecular size of the

surfactants can be compared by the value (1

𝛤𝑚𝑎𝑥) [6] and also, the value is biggest for

6

high-molecular size surfactants. As seen in the Figure 2, at the same fractional coverage

point, low-molecular size surfactant surface pressure value is greater than the high-

molecular size surfactant. Also, number of the low-molecular size surfactant is more than

the high-molecular size surfactant. In the graphic, in the moment when smaller molecules

reach the surface, surface pressure rise is observed. However, the effect of the biggest

molecules on the surface pressure starts after the molecular coverage reaches to a certain

value. The reaching time may cause an induction time to change of the surface tension.

Regime II forms by reduction of the surface tension. In the Regime II, the proteins

tend to conformation change at the surface, and it causes spaces between the proteins at

the surface. The spaces are filled from the bulk proteins. Thereby, protein concentration

is increased at the surface reducing the surface tension. Also, Regime II can be explained

that rigid parts of the adsorbed protein are relaxed by the conformational change and

desorption of the protein provides new interaction area at the surface [6].

7

After the surface is more saturated with the proteins, monolayer forms. Bulk

proteins can continue to adsorb to the monolayer forming multilayers. According to

Douillard and Lefebvre’s studies on the two-layer model of protein adsorption, surface

tension is affected only by first monolayer and second layer does not affect to the surface

tension [10]. Therefore, after the monolayer formation, the surface tension does not

change during multilayer formation resulting in constant surface tension of Regime III.

Prior studies exhibited small change in the surface tension; however, the change can

depend on the continuous small conformational change in the monolayer proteins [6].

1.3 Polymer Adsorption

Polymers can affect surface tension akin to proteins. However, since polymer

structure is different than protein, processes which polymer adsorption to the surface may

be different. Firstly, polymer adsorption process begins diffusion toward the surface. The

diffusion takes a certain time. After the polymer reaches the surface, the polymer

Figure 2 Surface pressure is a function of the fractional coverage using a Langmuir adsorption

model at 298 K [6]. Where, as 1/𝛤𝑚𝑎𝑥 = 20 Å2/𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 is low-molecular size surfactant and

1/𝛤𝑚𝑎𝑥= 2000 Å2/𝑚𝑜𝑙𝑒𝑐𝑢𝑙𝑒 is high-molecular size surfactant defined. Modified from

Beverung et al [6].

8

rearranges itself on the surface in time. Also during the diffusion and the rearrangement,

the polymer's desorption can occur to the bulk solvent. These processes result in

characteristic time dependent surface tension to reach what is formed the meso-

equilibrium point. According to Hua-Rosen study, the lag time is divided four regions

(region I is induction time, region II is rapid fall region, region III is meso-equilibrium

region, region IV is equilibrium region) [11]. The regions can be explained by diffusion

controlled adsorption kinetics [9, 12]. According to Ward and Tordai approach, the

relationship between time (t) and bulk polymer concentration (Cb) is 𝑡 ∝ 𝐶𝑏−2 for a certain

surface coverage [13].

1.4 Surfactants (surface active agent)

Figure 3 Surfactant is with a triple structure

Organic surfactants affect the surface tension when dissolved in water or an

aqueous solution. Also, surfactants usually decrease the surface tension. The surface

tension is basically Gibbs free energy in per unit area of surface. Surfactants are

amphiphilic which consists of both hydrophobic tail groups and hydrophilic head groups.

Hydrophobic molecules are non-polar molecules, and hydrophilic molecules are polar

molecules, and we know that water is a polar molecule. Therefore, according to the

9

aphorism “like dissolves like”, the hydrophobic tails tend to leave from aqueous solution

and the hydrophilic heads tend to go toward the aqueous solution. At the interfaces, these

tendencies provide the formation of the adsorption of surfactants. At the interface,

because of the adsorption of the surfactant molecules, intermolecular interaction forces

between water molecules increase and a diminution occurs in the surface tension of the

solution.

1.4.1 Anionic surfactants

If the polar head group is negatively charged, the surfactant is referred to as an

anionic surfactant. A hydrophobic group can be bonded to one or two hydrophilic groups,

such as sulfate, sulfonate, phosphate, and carboxylates alkyl sulfates. Anionic surfactants

are used more than the other kind of surfactants since their production is easier and

cheaper [14]. They are used in cleaning products, such as detergents, because solubility

is increased in the water and oil by anionic surfactants becoming counter-ion [14].

1.4.2 Cationic surfactant

If the polar head group is positively charged, the surfactant is called a cationic

surfactant. A hydrophobic group can be bonded to one or more hydrophilic groups. The

majority of cationic surfactant are based on the nitrogen atom carrying the cationic

charge [14]. Such as, alkyltrimethylammonium salts, cetylpyridinium chloride (CPC),

benzalkonium chloride (BAC), benzethonium chloride (BZT), 5-bromo-5-nitro-1,3-

dioxane, dimethyldioctadecylammonium chloride cetrimonium bromide, and diocta-

decyldimethylammonium bromide (DODAB). Cationic surfactants are used in surface

modifications such as softening, lubricating, corrosion inhibitors, and adhesion.

10

1.4.3 Nonionic surfactant

Surfactants without charge are usually called nonionic surfactants. The nonionic

part of the surfactant has a large number of mostly nitrogen, oxygen and sulfur atoms. In

contrast to ionic surfactants, physical properties of nonionic surfactants are not affected

by electrolytes significantly [14]. However, nonionic surfactants are affected by

temperature, and, in contrast to ionic surfactant, when temperature is increased; solubility

of nonionic surfactants is reduced becoming hydrophilic in water [14]. Polyoxyethylene

glycol alkyl ethers, polyoxypropylene glycol alkyl ethers, glucoside alkyl ethers, and

glycerol alkyl esters are the familiar example of the nonionic surfactants.

1.4.4 Zwitterion surfactant

If a surfactant has both positive and negative functional groups in the polar head

group, it is called as a zwitterionic surfactant, or an amphoteric surfactant. According to

their structure and ambient conditions, the surfactants may possess anionic and cationic

characteristics. Since they cause less damage to the skin and the eyes, they are used in

personal hygiene productions, such as hair shampoo, cleansing lotions, and liquid soaps.

The solution PH is important for these surfactants since it affects the surfactant charge,

and it can cause a change in physicochemical properties of the surfactants, such as

foaming, wetting properties, and cleaning effects. Zwitterion surfactants include the

surfactant examples, dodesil betain and cocamidopropyl betaine.

1.5 Micelle formation

A micelle form by surfactant molecules form cluster together. In solution, because

of their amphiphilic structure, surfactants change the solution physicochemical

properties, such as, changing surface tension of the solution [14]. Also, ionic surfactants

11

manner electrolyte in dilute solution, and in solution, increasing of the surfactant

concentration causes breaking down the delicate balance of electrostatic and hydration

interactions [14]. In aqueous solution, micelle formation is under the influence of two

forces [15], one of them is an attraction force causing molecules integration, and another

force is a repulsive force preventing unrestricted growth of the micelle size to become a

different macroscopic phase [15].

1.5.1 Thermodynamic of Micelle Formation

The formation of micelles can be explained by Gibbs free energy of mixing [16,

14]. The Gibbs free energy change is considered at constant temperature (T) and

pressure (P) [16]. As is known, surfactants are classified with respect to their polar head

group and due to their variations, Gibbs free energy would be different, each kind of

surfactant is examined in a separate way using different parameters.

According to Pseudo-Phase Separation Model, the chemical potential of the

monomer and surfactant in micelle form are equal at equilibrium [14]. The chemical

potential of the monomer and surfactant in micelle form are showed as µs and µm,

respectively [14].

µs = µm

The chemical potential of monomeric surfactants is given by the following equation:

µs = µs⃘ + RTlnxs

where µs⃘, chemical potential of the monomeric surfactant, is in the optimum state, xs,

mole fraction of monomer. Because the micelles are assumed to be in the optimum case

[14], µ⁰m = µm, and the Gibbs energy change resulting from the formation of micelles,

ΔGᵒmix is given as follow; Where α, micelles ionization degree

12

ΔGᵒmic = µ⁰m - µs⃘

= µm - µs + RTlnxs

=RTlnxs

The c.m.c value is equal to solubility limit of free monomers [14]. In this case, xs = xc.m.c,

and ΔGᵒmic is defined as follows;

ΔGᵒmic =RTlnxc.m.c

ΔGmix = ΔHmix – TΔSmix

where ΔHmix is enthalpy of mixing, ΔSmix is entropy of mixing and T is temperature.

Because of the surfactant structure, we have to discuss the surfactant energy changing in

different sections, such as tail and head group. The overall system has to be based; hence

we have to take the solvent energy changing into account due to Gibbs free energy of

mixing.

Consequently, c.m.c depends on the Gibbs free energy of mixing, and when total

Gibbs free energy of mixing becomes less than in the beginning condition of the Gibbs

free energy state, the surfactants will form micelle. Since the head groups always are

inside of the solution, there are small energy change observed due this, and it can be

considered negligible.

13

1.5.2 Parameters impacting c.m.c

1.5.2.1 Impact of Temperature and Pressure

The Krafft point is the minimum temperature at which solubility of the surfactant

is equal to c.m.c formation [16, 17]. Effect of the temperature is quite different on ionic

and non-ionic surfactants [16, 17]. For ionic surfactants, at the temperature below the

Krafft point, the surfactant solubility is significantly lower and micelles does not form

[16]. At temperature above the Krafft point, the surfactant solubility rapidly increases and

micelle formation occurs [16]. Temperature has the opposite effect on the non-ionic

surfactants [16]. When temperature is increased, the surfactants’ solubility is reduced and

the solution will become turbid at a point, which is referred to as cloud point [18, 16].

The solution will begin to phase separation [18, 16, 19].

1.5.2.2 Impact of Added Salt

Salt concentration affects the c.m.c formation especially for ionic surfactants [16,

20, 21]. Actually, the salt concentration effect is still small for non-ionic surfactants with

respect to ionic surfactants; however, the effect is significant [16]. The effect of the salt

concentration on c.m.c is demonstrated as follows [16];

log(c.m.c) = b2 + b3 C (non-ionic)

log(c.m.c) = b4 + b5 log C (ionic)

where, bj constants depend on the nature of the electrolyte. For the ionic surfactants,

when salt concentration is increased, the repulsive electrostatic force increases between

the head groups lowering the c.m.c [16, 14].

14

For ionic surfactants (E40-Foldon head groups have negative charge), when salt

concentration is increased, the effective head group size is reduced due to the decrease in

Debye length [16]. The shrinking in the head groups causes decreasing micelle surface

area compared to volume, which can lead to a changed shape of the micelle such as a

cylinder [16].

For nonionic surfactants, the salt acts like an electrolyte in the solution [7]. The

effects of the salt for nonionic surfactants are explained by the notion of ‘salting in’ and

‘salting out’ [16, 14, 22]. When the solution contains salt, the water molecules tend to

dissolve salt molecules [16]. However, the water molecules are needed to dissolve to the

hydrophilic part of the surfactants [16]. In the salted case, the water available for this is

reduced [16]. Therefore, the surfactants’ solubility and c.m.c. is reduced by salting out.

However, contain salts exhibit which is salting in the opposite behavior and c.m.c. is

increased [16].

1.5.2.3 Impact of Head Group and Chain Length

C.m.c is related to chain length of the surfactants, and the relationship is given by

the following equation [16, 14, 20];

log10c.m.c = b0 – b1mc

Where b0 and b1 are constant, mc is the number of carbon atoms in the chain for

surfactants which consists of hydrocarbon tails [16]. The previous studies show that the

nature of the head group can affect the value of b0 and b1, however, b1 is significantly

affected by the head group. Nonionic surfactants generally have larger b0 value than ionic

15

surfactants, but despite that, nonionic surfactant c.m.c. values are lower than ionic

surfactants [16]. Furthermore, variation of the hydrocarbon chain generally affects the

c.m.c. formation and this effect usually tends to increase the c.m.c [16, 14]. The variation

can be such as introduction branching, or double bonds, or polar functional group along

the chain [16].

1.5.2.4 Impact of Organic Molecules

Quite small amounts of organic molecules significantly affect c.m.c. [23, 14, 16],

and aqueous solutions of sodium dodecyl sulphate (SDS) can be given as a traditional

example for the effect [16]. In aqueous solutions, SDS causes reduction in surface tension

because of competing effects of adsorption of dodecanol at the air-water interface and in

the SDS micelles [16].

1.6 ELP –Foldon (MGH(GVGVPGEGVP)(GVGVP)41GWP-Foldon)

Elastin-like polypeptides (ELPs) consist of repeats of the pentapeptide (GβGαP)n,

and α which is in the parenthesis can be any of the 20 naturally occurring amino acids, β

can be any of those amino acids except for proline, n is the repeated number of the

monomer [19]. Since the side chain of the proline is bonded covalently to the nitrogen

atom of the peptide backbone, it does not have amide hydrogen to use as a donor in

hydrogen bonding [5]. An important characteristic of these polypeptides is their LCST

(lower critical solution temperature) behavior, which are thermally responsive polymers

exhibit phase separation above a transition temperature (Tt) [19]. The polymer is soluble

in water below Tt, and when the temperature is increased the polymer shows aggregation

[19].

16

Figure 4 The figure shows the polymer’s arms aggregation with respect to temperature [19].

Above Tt, the coacervate phase, which is viscoelastic and dense, is formed.

Coacervation process is reversible, and two components which are water and

polypentapeptide form at above Tt. Below the Tt, the polymer (ELP) is going to be

soluble in the water and does not show surfactant properties. And the polymers disperse

as a homogeneous in the solution. However, this case can cause changing the surface

tension since the solvent is not becoming pure.

Above Tt, the polymer is insoluble in the water and shows surfactants properties.

The trimer which is aggregated at elevated temperatures gains hydrophobic features. In

this case, the polymer going to toward solvent surface, polymer tail which is trimer is

located air and head group of the polymer is located at the solvent surface. When the

surfactants concentration reaches the c.m.c point, the polymers form micelles in the

solution.

1.7 Thermodynamics of ELP-foldon

ELP-foldon consists of three MGH(GVGVPGEGVP)(GVGVP)41GWP-

GYIPEAPRDGQAYVRKDGEWVLLSTFL polymers, which are held together though the

17

trimer forming ELP-foldon head group (foldon) [23]. The ELP chains form the tails, and

because of their three tails, they are referred to a three-armed star polypeptide [23, 18].

Since the ELP-foldon tails show hydrophobic properties at elevated temperatures and the

head group shows hydrophilic properties, ELP-foldon can be used as a surfactant in

liquid [23]. Because of ELP’s thermally responsive features, by increasing temperature,

the arms undergo conformational changes and they encourage micelle formations [23].

Prior studies show that micelle formations are observed in low salt above the transition

temperature (Tt) of the ELP. However, at physiological salt concentrations above the Tt,

turbidity, which is measured with UV-vis spectroscopy occurs [18]. In ELP-foldon

system, micelle formations depend on the system pH, salt concentration, polarity, and the

ELP molecular length and size [1, 16]. The molecular length and size also affect viscosity

of the ELP [1]. Since larger molecules interact with liquid molecules more, the molecular

movement becomes slower, and that causes diffusion of the molecules to be slower. In

addition, as observed on the previous studies, polymers which have the biggest molecular

size affect surface tension of the system when the polymers reach a certain coverage

number in the surface [6]. Thereby, occurrence of the ELP adsorption on the surface is

time dependent, related to the diffusion coefficient, resulting in time variation between

different molecular size ELPs for adsorption of the ELP on the surface. Furthermore,

since temperature affects diffusion velocity, time variation can be observed at different

temperatures.

18

In Figure 5, ELP-foldon polymer surfactants are illustrated in three states. Each

state is explained below using general thermodynamic properties. Notably, since the head

groups and the tails have different chemical properties, they will be examined separately.

Sum of head group contributions to Gibbs free energy of mixing is close to zero, so it is

assumed to be negligible. Changes of the enthalpy are small between the each state since

the intermolecular bonds are not changed. However, the changes of the orientation of the

molecules and hydrophobic effects cause changes of entropy which dominant the process.

In State 1, the surfactant tails and head groups are in solution. Entropy (ΔS) of

the water molecules which are closer to the tail will be smaller due to the water

molecules’ order which is increased. Water molecules are bonded to each other by

hydrogen bonds. When the surfactant is located in the water, the surfactant takes the

water molecule's place. Thereby, water molecules lose H-bonds and they prefer to gain

the bonds again. Otherwise, due to the tail being hydrophobic, water molecules do not H-

bond with the tail. In this case, the water molecules are going to bond to each other but,

firstly, the water molecules have to be in the appropriate position, and hence they need to

Figure 5 The surfactant entropy change is explained for each section. 1) In the water, the surfactants

are in the free form. 2) The surfactants are in the aqueous solution-air interface. 3) The surfactants are

in the micelle form.

Air

Water

1.

2.

3.

19

be reoriented (conformational change) to bond to each other. The event causes increasing

order (decreasing of entropy) in the solvent (water). This is called the hydrophobic effect.

Moreover, enthalpy (ΔH) of the solvent molecules which is closer to the tail is increased.

Furthermore, the tail of the molecule is located in the bulk, so disorder (entropy) and

enthalpy is greater than when located on the surface, since the bulk surfactant has more

orientation state than to be located on the surface of the solvent.

The water molecules interact with the head group due to the head group being

hydrophilic. Thereby, the head group disorder entropy is increased compared to the

beginning conditions. Enthalpy is structural stored energy of the matter. And it is

described as the sum of the internal energy and potential energy of the matter. Therefore,

enthalpy depends on the molecular bonding energy.

In State 2, the hydrophobic tails of the surfactants are located in the air and

hydrophilic head group is located in the solution (Figure 5). The surface tension is

reduced because the head groups interact to satisfy the lack a bonding compared to bulk

water molecules. Also, the surface molecules because disordered by interacting with the

surfactant. Compared State 1, the water molecules that were surrounding the tail will be

released to increase disorder, resulting in higher entropy. However, the tail is confined at,

the interface decreasing its entropy. The surface molecules have more order (less

entropy), but the overall system has less order (more entropy) because of the disordering

of the water.

The surfactants assemble as micelle in State 3. There is limited surface available

for the surfactants to occupy State 2 and reduce their energy. The concentration at which

surface saturation is reached is referred to as the critical micelle concentration (c.m.c).

20

The formation of micelles is similar to State 2 in that the hydrophobic tails are separated

from the water by the hydrophilic head group the molecules become slightly more order

than in State 2. It leads to reduce the entropy with respect to surfactants which are in the

surface. Since the surface reaches the maximum capacity, when surfactant concentration

is increased in the system, the surface tension will be constant. In other words, surface

tension is not changed by concentration, and the added surfactants form micelle in the

system.

In this study, at different temperatures and different concentrations, the effect of

ELP on the surface tension is investigated. Also, according to the ELP’s structure,

diffusion of the ELP to the surface and the diffusion’s dependency is viewed.

Furthermore, c.m.c of the ELP-foldon and its dependency of temperature are

investigated.

Table 1.1 The table shows sum of the free energy change of the system.

S H Tail Head Tail Head

Solvent Molecule Solvent Molecule Solvent Molecule Solvent Molecule

1 max Low high Ø high high high Ø Ø

2 surface High low Ø low low low Ø Ø

3 micelle High low Ø low low low Ø Ø

For

micelle

Favorable Unfavorable Unfavorable Favorable Favorable

21

CHAPTER II

MATERIALS AND METHOD

2.1 Expression and Purification of ELP-Foldon

The protein-based polymer surfactant used in these experiments is ELP-foldon

(MGH(GVGVPGEGVP)(GVGVP)41GWP-GYIPEAPRDGQAYVRKDGEWVLLSTFL).

It consist of 43 pentapeptide repeats, all of them except one are GVGVP. One of the N-

terminal pentapeptides is GEGVP. The substitution of the glutamic acid introduces a

negative charge at neutral pH to counteract the positive charge of the N-terminus. This

allows micelle formation at neutral pH. The ELP-foldon is produced in an E. coli

expression system. Cultures are prepared by adding a small sample from a frozen

bacterial stock to 10 ml Luria Broth (LB) with 100 μg/mL ampicillin. The culture is left

overnight in an incubator which is shaking at 37 ℃. LB culture medium is prepared by

adding 10 g peptone, 5 g NaCl, and 5 g yeast extract to 1 L purified water. The medium is

put in the autoclave at 121 ℃ for around 60 minutes. After cooling, 100 μg/mL ampicillin

is added to the medium. A 1 ml sample is taken from the medium to generate a reference

point for optical density (O.D). Then, the overnight culture is transferred to the medium.

Until the OD has reached a desirable point which is around 1.0, the medium is kept in the

22

incubator which is shaking at 37 ℃. After this point, the bacteria number has reached a

desirable number to induce expression by the addition of 0.24 g/l isopropyl β-D-1-

thiogalactopyranoside (IPTG). The culture is kept 4-5 hours in the incubator which is

shaking at 37 ℃. To harvest the cells, the culture is centrifuged 20 minutes at 2-3 ⁰С,

14000 xg to obtain pelleted bacteria. To purify the ELP from the bacteria, the pellet is

resuspended adding phosphate buffered saline (PBS) and the cells are lysed by

sonication. Centrifugation is carried out both cold and hot to utilize the thermally

responsive behavior of the ELP to purify it. Cold centrifugation is performed 20 minutes

at 2-3 ⁰С and 20400 xg and hot centrifugation processes is performed 20 minutes, at 43-

45 ⁰С and 7700 xg. After the sonication process, the centrifugation process steps are first

cold centrifugation, first hot centrifugation, second cold centrifugation, second hot

centrifugation, and third cold centrifugation, resulting with the final protein in the

supernatant.

The concentration of the final protein is measured by UV-light absorption at 280

nm using a spectrophotometer. To convert absorbance to concentration, Beer's law is

used. Aromatic side chains (tryptophan (W), tyrosine (Y), and cysteine (C)) absorb the

UV-light [24]. The absorbance and the concentration are related linearly, through an

extinction coefficient as expressed by Beer’s law [24]. The extinction coefficient for the

ELP-foldon is 13980 M-1

cm-1

.

SDS-PAGE (sodium dodecyl sulfate-polyacrylamide gel electrophoresis) is used

to verify molecular weight and purity. For the SDS-PAGE process, 15 𝜇𝑙 protein sample,

3 𝜇𝑙 6 × 𝑑𝑦𝑒, and 5 𝜇𝑙 marker are used. To observe the trimer formation of the protein,

the solution is either boiled or unboiled prior to addition to the gel. The gel is submersed

23

in buffer solution applying 100 V electrical voltages. After an hour, the gel was removed

from the buffer solution, rinsed, and stained with coomassie blue.

2.2 Surface Tension Apparatus

To measure surface tension, several methods are available in the literature such as

maximum pull on a rod (Du Noüy-Padday), Wilhelmy plate, Du Noüy ring, spinning

drop, bubble pressure, and drop shape methods. We used a pendant drop shape methods.

The methods, which do not depend on the contact angle [25], are based on the analysis of

drop shape which is obtained from the shape of a sessile drop, pendant drop or captive

bubble to determine the liquid–vapor or liquid–liquid interfacial tensions. The shape of a

drop is determined by a combination of surface tension and gravity effects [26]. Drop

shape methods can be used in many difficult experimental conditions since they have a

lot of advantages in comparison to the other techniques [26].

To obtain pendant drop image, a Ramѐ-Hart Goniometer/Tensiometer is used.

The tensiometer consists of temperature controller, CCD camera (home built CCD

camera with computer capture is used), fiber optic light source, environmental chamber,

chamber cover with stage, elevated temperature syringe, glass syringe, stainless steel

needle, film clamps, microsyringe fixture, and base. The temperature controllers provide

accurate temperature control on the environmental chamber and the elevated temperature

syringe. The environmental chamber also protects the drop from adverse effects. The

glass syringe is assembled into the elevated temperature syringe that keeps the glass

syringe and its contents at a controlled temperature. The drop is formed by an adjustable

screw of the apparatus applying pressure to the plunger. The melting point of the

materials is up to 230 ℃ [27], although we made measurements only up to 75 ℃. The

24

chamber cover can be tilted to align the sample. The microsyringe fixture holds an

elevated temperature syringe, and it can be adjusted in all directions. All of the

components are assembled on the base. Details about the CCD camera, the fiber optic

light source, and the needle are described in the image analysis section 2.3.2. The

schematic diagram of the parts is illustrated in Figure 7.

2.3 Axisymmetric Drop Shape Analysis (ADSA)

Surface tension determination by axisymmetric drop shape analysis (ADSA) was

first introduced by Bashforth and Adams and it continues to this day [28, 29]. A second

generation of ADSA was developed by del Río [29, 30, 31] using the curvature at the

apex rather than the radius of curvature and the angle of vertical alignment as

optimization parameters [26]. A flowchart (Figure 6) shows the general procedure of

ADSA to measure surface tension. ADSA uses drop interface properties which are

obtained from the shape of pendant drops or sessile drops found by analyzing the images.

The coordinate profile properties (i.e. the experimental profile) of the drops are obtained

for use in numerical optimization processing. After that, the experimental properties of

the drop and physical properties such as density and gravity are used to fit a series of

Laplacian curves to obtain liquid–fluid interfacial tension, contact angle (in the case of

sessile drops), drop volume, surface area, radius of curvature at the apex, and the radius

of the contact circle between the liquid and solid (in the case of sessile drops) [26]. The

images were analyzed using MatLab® codes. The codes were written by Eric Helm

loosely based on code found in literature [25].

25

Figure 6 General procedure of Axisymmetric Drop Shape Analysis (ADSA).

2.3.1 Image Capture

Figure 7 Schematic diagram of the experimental setup of ADSA for analysis of pendant drop. Modified M.

Hoorfar et al. [26].

26

To analyze a drop image, the image is obtained from the experimental setup of

ADSA shown in Figure 7. A fiber optic light source is used behind the drop to improve

image contrast. A Vivicam 3750 CCD camera was used to obtain images. Each pixel of

the images consists of bits which describe gray scale. Mathematically, the relationship

between expressions is showed in the following equation:

𝐿 = 2𝑘

where L is the number of gray levels or the shades of gray and k is bits per pixel (bpp).

The camera resolution is 2048 × 1536 pixels and 24 bits hence gray levels is 224 or

16,777,216 different shades. The picture is analyzed and the appropriate parameters are

solved based on the Young-Laplace equation.

2.3.2 Image Analysis

2.3.2.1 Edge Detection

The image analysis process first begins by the software edge detection procedure

[26, 32, 33], consisting of three steps (Figure 8) [26]. The pendant drop image is loaded

into MatLab® as an original image (Figure 9A). To improve visibility of the drop in the

Figure 8 The process used to obtain the final drop

profile. Modified from M. Hoorfar et al. [26].

Edge Detection (Sobel, Canny)

Correction of Optical Distortion

Correction for the Misalignment of Camera

Final Drop Profile

Drop Image

27

image, the image is changed to grayscale then binary using a threshold. By this method,

an image consisting of black and white pixels is obtained (Figure 9B).

The original image contains noise and useless information, hence to reduce the

noise and useless information preserving important information, filter and edge detection

Figure 9 Drop image for steps of image analysis process A) The drop image of 1 μM solution of ELP-

foldon is at pH of 7.4, 10th minute, and a salt concentration of 25 mM. B) Binary image is obtained

using threshold. C) The image is after edge detection. It is obtained using Canny operator. D) Final

drop profile image is obtained after boundary trace process.

A.

. B.

C. D.

28

operators are applied to the image [34]. All edges are detected using the MatLab® edge

detectors, Canny [26]. Figure 9C shows edge points after the edge detection operator is

applied on the drop image.

Sobel edge detection was also attempted. It is one of the most well-known image

processing algorithms [26, 34]. Two convolution kernel algorithms (3 × 3) are used in

the Sobel [26, 34]. While one of them is used to find the horizontal edges, another one is

used to find vertical edges. Basically, these two kernels are perpendicular to each other

(Figure 10) [34]. These kernels help to determine sudden light intensity change within the

image.

Basically, the image is divided into (3×3) pixels in size, and then gradients are

calculated applying the kernels separately to the image [34]. The gradients are combined

together to describe exact gradient magnitude [26, 34]. Gradient magnitude is calculated

using the following equation:

|𝐺| = √𝐺𝑥2 + 𝐺𝑦

2

The gradient direction is calculated by:

𝜃 = 𝑎𝑟𝑐𝑡𝑎𝑛 (𝐺𝑦

𝐺𝑥)

𝐺𝑥 = [−1 0 +1−2 0 +2−1 0 +1

] 𝐺𝑦 = [+1 +2 +1 0 0 0−1 −2 −1

]

Figure 10 The kernels’ horizontal (𝑮𝒙) and vertical (𝑮𝒚) derivative

approximations [34].

29

According to studies, the Canny edge detection technique performs better

compared to other techniques for surface tension measurement [26, 34] therefore, we

used the Canny edge detection technique. The Canny uses Gaussian filter to eliminate the

noise and useless information from the original image using standard convolution

methods [26, 34]. Thereby, a smoothed image which is purified from noise is obtained.

The gradient intensity and direction are computed for the image [26, 34]. Non-maximum

suppression is applied to reduce thick edge responses to thin lines [34, 35]. Hysteresis is

used to determine beginning and end points of the edge using two different threshold

values, a high and a low [26, 34].

In our diagram, a high resolution image is used. In the high resolution image, the

intermediate values exist because of pixel density. Thereby, boundary trace methods are

applied directly to the image to obtain drop profile, which is illustrated in Figure 9.D.

Since the unit of the experimental edge value is in pixels, it needs to be converted

to millimeters to compute the theoretical Laplace equation [26]. A calibration grid was

used to verify this procedure to convert pixels to millimeters. In this process, the diameter

of the syringe needle is used as a reference to determine the ratio of millimeters to pixels.

The needle diameter was measured as 0.72 mm. The number of pixels across the needle

is determined for a row of pixel, yielding the number of pixels for 0.72 mm, the process

is applied for 20 rows of pixels, and the average is taken to calculate the conversion to

millimeters.

Once the capillary diameter is calculated, the Z-axes height cut-off point is

selected. The number of analyzed points is changed with respect to the cutoff point.

Thereby, the drop curve, which will be analyzed, is obtained (Figure 11). In our study,

30

the cut-off point level was selected close to the needle, and when different cut-off points

were selected close to the needle, the variability of the surface tension value was

insignificant.

To observe and correct for the camera and experimental setup errors, vertical

symmetric axis of a pendent drop image curve is found, and a midpoint line is formed

based on the midpoint of several horizontal values (Figure 12A). The red line which is

shown in Figure 12B. is formed basing each horizontal pixel point coordinate of the drop

edge as vertical symmetrical axis of a pendent drop image curve. The profile is divided

into two parts by the midpoint line. However, if the camera and drop were not in

alignment, the midpoint line would not divide the drop curve symmetrically. In other

words, the coordinates of pixel points (𝑋𝑖, 𝑍𝑖) on the same line are not exactly equidistant

Figure 11 The red line is shown drop profile after cut-off

point is selected.

31

from the midpoint line. When the blue line is not exactly described by the software, it

means that the edge coordinate points are not described well and its reason can be light

errors or camera’s sharpness. If a red line forms a curve close to the holder, the error is

caused from the curvature of the camera.

a. b.

Figure 12 a) The blue line is midpoint of drop profile b) The red line is midpoint of each curve point.

32

2.3.3 Drop Shape Calculation

The balance between surface tension and external forces is described

mathematically using a set of initial parameters which are fit to the drop profile [26] by

the Young-Laplace equation of capillarity,

𝛥𝛲 = 𝛾 (1

𝑅1+

1

𝑅2) (1)

where R1 and R2 are the two principal radii of curvature, 𝛾 is surface tension of the drop,

and ΔP is the pressure difference across the liquid interface. If there is not any external

force except gravity in the surrounding environment of the drop, ΔP can be expressed as

a linear function of the elevation:

∆𝑃 = ∆𝑃0 + (∆𝜌)𝑔𝑧 (2)

Figure 13 The image shows the distribution

of the parameters geometrically on the drop

[26, 52].

+𝑍

+𝑋

The coordinate system shows

direction of axises.

33

where ΔP0 is the pressure difference at a reference plane and z is the vertical coordinate

of the drop measured from the reference plane. Also, when the value of 𝛾 is given, by

inverse calculation, the shape of the drop can be determined [30].

∆𝑃0 + (∆𝜌)𝑔𝑧 = 𝛾 (1

𝑅1+

1

𝑅2) (3)

Two principal radii are determined by two planes which are defined at any point

of a curved surface (𝑋𝑖, 𝑍𝑖) [26]. One of the planes passes through the surface, and a

curve is generated between the plane and the surface containing a normal [26], thereby,

the first radius of curvature is generated [26]. To describe the second radius of the

curvature, another plane is passed through the surface being perpendicular to the first

plane [26]. Under the assumption of axial-symmetry (between the interface and z-axis),

the principal radius of curvature, R1, is related to the arc length, s, and the angle of

inclination of the interface to the horizontal, 𝛷, by [26, 30, 36, 37]

1

𝑅1=

𝑑𝛷

𝑑𝑠 (4)

The second radius of curvature is given by [26]

1

𝑅2=

sin 𝛷

𝑥 (5)

Figure 13 represents the ADSA coordinate system. In this system, “mean curvature” is

described by summing (1

𝑅1+

1

𝑅2) of two principal radii of curvature [26]. The drop’s apex

34

curvature is defined as “b” [31], and because of the axial-symmetry, at the apex, the b

value is constant in all directions and the two principal radii of curvature are equal, i.e.,

1

𝑅1=

1

𝑅2=

1

𝑅0= 𝑏 (6)

where, R0 is the radius of curvature [26]. At the apex, the arc length, s, is equal to zero

[26]. Thereby, in this point, the pressure difference is expressed using equation 1 as [26]

𝑃0 =2𝛾

𝑏 (7)

The following boundary-value problem is obtained as a function of the functions of the

arc length, s, using equation 4, 5 and 7 into equation 1 [26]

𝑑𝛷

𝑑𝑠=

2

𝑏+ 𝑐𝑧 −

sin 𝛷

𝑥 (8)

𝑐 =(∆𝜌)𝑔

𝛾 (9)

where c is a capillary constant, and because the gravity, g, has positive values for sessile

drops and negative values for pendant drops [26].

Equation (8) together with the geometrical relations [26]

𝑑𝑥

𝑑𝑠= cos 𝛷 (10)

𝑑𝑧

𝑑𝑠= sin 𝛷 (11)

35

form a set of first order differential equations for x, z, and ϕ as functions of the arc length,

s, with the boundary conditions [26]

𝑥(0) = 𝑧(0) = 𝛷(0) = 0 (12)

Also, at s=0

𝑑𝛷

𝑑𝑠= 𝑏 (13)

The Laplacian axisymmetric fluid–liquid interface curve was generated by

solving these equations numerically [25, 26] using a Runge-Kutta method [36, 38, 39, 40,

41] for given values of b and c [26]. We programmed this using the ODE45 function

in MatLab®. Dimensionless parameters were substituted into Equation 8. The values are

normalized using apex curvature of the drop, b [25]:

𝑥 =𝑥

𝑏

𝑠 =𝑠

𝑏 (14)

𝑧 =𝑧

𝑏

geometric consideration [25]

𝑑𝛷

𝑑𝑠= 2 + 𝑐𝑧𝒃𝟐 −

sin 𝛷

𝑥 (15)

𝑑𝑥

𝑑𝑠= cos 𝛷 (16)

𝑑𝑧

𝑑𝑠= sin 𝛷 (17)

36

The initial conditions [25],

𝑥(0) = 𝑧(0) = 𝛷(0) = 0 (18)

A theoretical curve generated using the Young-Laplace equation is illustrated in Figure

14.

2.3.4 Optimization

The values b and c are defined from the experimental profile, and these values are

used in the Laplace equation to generate a theoretical profile. The two profiles are

mapped and errors are found using error function, 𝑒𝑖. For each experimental data point

(𝑋𝑖, 𝑍𝑖), the closest to the theoretical curve point (𝑥𝑖, 𝑧𝑖) is selected, and the distance

Figure 14 Drop profile analysis is used for the fitting process. The red

curve is a theoretical curve and is generated from the Young-Laplace

equation, the black is an experimental curve and is obtained from the

picture. The green line is a rotated curve with respect to the original image.

37

between these points, 𝑑𝑖, is calculated [26]. The error function is defined as 𝑒𝑖 =1

2𝑑𝑖

2 [26,

29, 42].

𝑒𝑖 = 𝑑𝑖2 = (𝑥𝑖 − 𝑋𝑖)

2 + (𝑧𝑖 − 𝑍𝑖)2 (19)

The fitting process minimizes the objective function, E, which is described as the

sum of the individual errors squared. The function contains the fitting parameter, q, with

elements qk, k=1, ⋯, M. Best fit between the experimental points and a Laplacian curve

is obtained finding q values that minimize E. A point is necessary in order to calculate the

objective function and is assumed a minimum value at the point in the maximum M

value. The objective function is a defined function of a set of parameters at following.

𝜕𝐸

𝜕𝑞𝑘= ∑

𝜕𝑒𝑖

𝜕𝑞𝑘

𝑁𝑖=1 = 0, 𝑘 = 1, … . , 𝑀 (20)

The objective function consists of nonlinear algebraic equations, hence an iterative

solution is required using numerical solver such as Newton–Raphson method,

Levenberg–Marquardt method and Nelder-Mead simplex method [26, 36]. While first

generation of ADSA uses Newton–Raphson method, second generation uses Newton–

Raphson method and Levenberg–Marquardt method together [26, 36]. In our program,

we used the Nelder-Mead method as a numerical solver using MatLab®. Figure 15 shows

the final drop curve of after the optimization processes and the residuals of the fit.

Nelder-Mead is a simplex method that is used to find a local minimum point of a few

variable functions [43]. For two variables, the simplex forms a triangle and it is a method

38

of comparing the value of the functions in the three vertices of the triangle. The function,

where value is the largest peak value is rejected and a new peak value is determined

Thereby, a new triangle is created and the process is continued. The coordinates of the

minimum points are found reducing the size of the triangle. The algorithm is created

using the simplex term, and the minimum point of the function of N variables will be

found by this algorithm in N dimensions.

At the minimum error point, the value of b and c are determined. A graph

illustrating error as a function of c and b gives an idea of the error sensitivity to the

determined values (Figure 16). Surface tension is calculated substituting c, g and ∆𝜌

values into Equation 9. The obtained results for this example are shown in Table 2.1.

a. b.

Figure 15 a) The final drop curve results, theoretical and experimental curve are overlapped by optimization processes.

b) The residual plot is formed by remaining from the difference between the theoretical and the experimental curve.

39

Surface tension was measured at different polymer concentrations (from 10 nm to

50 μM) and temperature (from 25 to 35℃). Before we used the ELP-Foldon solution, the

solution was filtrated. Each concentration was prepared from highest concentration to

lower concentration and the concentration was measured using UV-spectroscopy at 280

c (mm-2

) b (mm) Error Surface Tension

(mN/m)

−0.194 ± 0.003 1.1306 ± 0.016 0.0007 ± 0.001 50.6 ± 1.5

Table 2.1 The sample image of 1 μM solution of ELP-foldon at pH 7.4, at 10 min, and a salt

concentration of 25 mM. Approximate values (±) are based on experimental pure water

surface tension compared with literature surface tension values of water.

Figure 16 The minimum value of x against the value of b and c are described.

The sample image parameters are determined: b= 1.1306 mm and c=-0.1937

mm-2

.

40

nm to obtain precise concentration values. The solutions were taken from the solution

tube using a glass syringe. After the syringe was assembled to the tensiometer, 5 minutes

elapsed for solution to reach desired temperature. The tensiometer and the camera were

calibrated taking sample drop pictures and analyzing with the MatLab® code. To reduce

evaporation from the drop, the humidity in the chamber was increased by placing two

drops of solution on the chamber surface. An experimental drop was created immediately

after a previous drop was dropped and timer was started. To obtain pictures, the camera

utilized auto-shooting mode with a 10 second delay to avoid vibrations. During each

drop, a picture was taken every 30 seconds. For each concentration, at the same

temperature, the experiment was repeated at least three times.

In order to validate the experimental method, experimental surface tension values

of water were compared to values reported by N.B. Vargaftik et al [44] (Figure 17). As is

seen from the graph, the experimental values and literature values are close to each other

up to 55 ℃. For the experiment, drop images were taken by the camera in 10 ℃

increments. The experimental water surface tension value is obtained by averages of the

three images at each temperature point.

41

Figure 17 The graph shows a comparison of literature values of surface tension of water and

experimental values of water as function of the temperature.

40

45

50

55

60

65

70

75

80

85

90

20 30 40 50 60 70 80 90

Experimental water

Literature Water

Temperature (℃)

Surf

ace

Te

nsi

on

(m

N/m

)

42

CHAPTER III

RESULTS AND DISCUSSION

The surface tension as a function of temperature was measured for seven different

concentrations. The behavior varied with concentration resulting in three general cases

for lower, intermediate, and higher concentrations.

At the lower concentrations, (≤ 100 𝑛𝑀) the surface tension was not affected by

the protein within the time of the experiment. The samples were measured at times as

long as 2 hours with no observed change in the surface tension. It is possible that there

was not sufficient time for the protein to diffuse and create a monolayer at the surface.

The change of surface tension observed as a function of temperature is equivalent to the

surface tension change of the pure solution (Figure 18).

43

At intermediate concentrations, between 0.2 and 1.0 µM, the surface tension

Figure 18 Measured surface tension as a function of temperature for solutions of different

concentrations of ELP-foldon at pH 7.4 and a salt concentration of 25 mM PBS. A) 10 nM, B) 31.6 nM,

and C) 0.1 μM.

40

45

50

55

60

65

70

75

80

20 30 40 50 60 70

10 nM

10 nM

Temperature (℃)

Surf

ace

Ten

sio

n (

mN

/m)

A.

40

45

50

55

60

65

70

75

80

20 30 40 50 60 70

31.6 nM

Surf

ace

Ten

sio

n (

mN

/m)

Temperature (℃)

31.6 nMB.

40

45

50

55

60

65

70

75

80

20 30 40 50 60 70

0.1 µM

0.1 µM

Surf

ace

Ten

sio

n (

mN

/m)

Temperature (℃)

C.

44

varied with time and temperature (Figure 19A, 20A, 21A, 22A). The behavior is similar

to what was observed by Hua-Rosen for surfactant adsorption. There was an initial period

of small decrease in the surface tension followed by more rapid decrease to a more stable

meso-equilibrium value. The initial and meso-equilibrium values varied with temperature

comparable to the solvent value (Figure 19, 20, 21, 22) and the times required to reach

the meso-equilibrium decreased with increased temperature and concentration.

For example, Figure 19A shows that at 0.2 𝜇𝑀 polymer concentration, the surface

tension change is a function of time and temperature. At 25 ℃, for the first the 10

minutes, the surface tension was approximately equal to the solvent surface tension

(~72 𝑚𝑁/𝑚). A surface tension decrease was observed after approximately 10 minutes,

and the decrease continued until about 23 minutes. After this, change was not observed.

At this time, the surface tension was ~50 ± 1.5 𝑚𝑁/𝑚. At 30 ℃, a drop in surface

tension began in less than 10 minutes. The reduction continued between 10 and 20

minutes, after which, change was not observed. At this time, the surface tension was

determined as ~50 ± 1.5 𝑚𝑁/𝑚. At 35 ℃, the surface tension decrease began at the

fourth minute and continued until 18 minutes when the surface tension remained constant

at ~49 ± 1.5 𝑚𝑁/𝑚.

45

Figure 19 A) Measured surface tension as a function of time for a 0.2 μM solution of ELP-

foldon at pH of 7.4 and a salt concentration of 25 mM PBS at different temperatures. B)

Measured surface tension as a function of temperature for a 0.2 μM solution of ELP-foldon

at pH of 7.4 and a salt concentration of 25 mM.

30

40

50

60

70

80

90

1 1.5 2 2.5 3 3.5 4

0.2 µM

35 ℃

30 ℃

25 ℃

Surf

ace

Te

nsi

on

(m

N/m

)

log(time (s))

A.

0

10

20

30

40

50

60

70

80

20 25 30 35 40

0.2 µM

0.2 µM

Surf

ace

Ten

sio

n (

mN

/m)

Temperature (℃)

B.

46

Figure 20 A) Measured surface tension as a function of time for a 0.316 μM solution of

ELP-foldon at pH of 7.4 and a salt concentration of 25 mM PBS at different temperatures.

B) Measured surface tension as a function of temperature for a 0.316 μM solution of ELP-

foldon at pH of 7.4 and a salt concentration of 25 mM.

45

50

55

60

65

70

75

1 1.5 2 2.5 3 3.5

0.316 µM

25 ℃

27.5 ℃

30 ℃

32.5 ℃

35 ℃

Surf

ace

Te

nsi

on

(m

N/m

)

A.

0

10

20

30

40

50

60

70

80

20 25 30 35 40

0.316 µM

0.316 µM

Surf

ace

Ten

sio

n (

mN

/m)

Temperature (℃)

B.

log(time (s))

47

Figure 21 A) Measured surface tension as a function of time for a 0.1 μM solution of ELP-foldon at

pH of 7.4 and a salt concentration of 25 mM PBS at different temperatures. B) Measured surface

tension as a function of temperature for a 0.1 μM solution of ELP-foldon at pH of 7.4 and a salt

concentration of 25 mM.

30

40

50

60

70

80

90

1 1.5 2 2.5 3

1 µM

27.5

25 ℃

30 ℃

35 ℃

Surf

ace

Te

nsi

on

(m

N/m

)

log(time (s))

A.

0

10

20

30

40

50

60

70

80

20 25 30 35 40

1 µM

1 µM

Surf

ace

Ten

sio

n (

mN

/m)

Temperature (℃)

B.

48

At higher concentration (50 𝜇𝑀), the surface tension had reached meso-equilibrium at

the first time point measured for all temperatures. The meso-equilibrium surface tension

decreases with temperature comparable to the solvent as is indicate by a constant surface

pressure (Table 3.1)

Figure 22A was obtained at 50 𝜇𝑀 polymer concentration, at pH of 7.4 and a salt

concentration of 15 𝑚𝑀 PBS as a function of time and temperature. The surface tension

reached meso-equilibrium quickly between 25 and 50 ℃. At this concentration, micelle

formation has been observed at 50 ℃ as shown in Figure 22B, yet no change in surface

pressure is observed. The solution and the solvent surface tension values are presented in

Table 3.1.

The surface pressure is defined as 𝛱 = 𝛾0 − 𝛾, where, 𝛱 is surface pressure, 𝛾0 is

solvent surface tension and 𝛾 is the solution surface tension. The difference between the

solution surface tension and the solvent surface tension has been approximately constant

for the temperatures studied resulting in a constant surface pressure (Figure 23).

49

Figure 22 A) Measured surface tension is a function of time for a 50 μM solution of ELP-foldon at pH

of 7.4 and a salt concentration of 15 mM PBS at different temperatures. B) Measured surface tension

and rate of absorption (UV) is a function of temperature for a 50 μM solution of ELP-foldon at pH of

7.4 and a salt concentration of 15 mM.

20

25

30

35

40

45

50

55

60

65

70

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6

25 ℃

30 ℃

35 ℃

40 ℃

45 ℃

50 ℃

50 µM

Surf

ace

Te

nsi

on

(m

N/m

)

log(time (s))

A.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

30

35

40

45

50

55

60

65

70

20 25 30 35 40 45 50 55 60 65 70

Surface Tension

Absorption

Surf

ace

Ten

sio

n (

mN

/m)

Temperature (℃)

B. 50 µM

Ab

sorb

ance

@ 3

50

nm

50

Table 3.1 The solution and the solvent surface tension values and surface pressure values

at different temperatures for 50 µM.

Temperature

(℃)

The solution Surface

tension (𝒎𝑵/𝒎)

The solvent Surface

tension (𝒎𝑵/𝒎)

The surface

pressure (𝛾0 − 𝛾)

(𝒎𝑵/𝒎)

25 50.5 ±1.5 72 21.5

30 49 ±1.5 71.2 22.2

35 48.5 ±1.5 70.4 21.8

40 47.8 ±1.5 69.6 21.8

45 47.2 ±1.5 68.8 21.6

50 46.3±1.5 67.9 21.6

Critical micelle concentration (c.m.c) is a point in which the surface tension is

stable and does not change when the concentration and temperature are increased (Table

3.2). For a 50 μM, measurement of rate of absorption (UV) shows micelle formation after

40 ℃ (Figure 22). Despite micelle formed, surface tension does not change.

Figure 23 The surface pressure as a function of the temperature at 50 μM polymer concentration, at pH

of 7.4 and a salt concentration of 15 mM PBS.

0

5

10

15

20

25

30

35

40

20 25 30 35 40 45 50 55

Temperature (℃)

50 µM

γₒ- γ

The

su

rfac

e p

ress

ure

(m

N/m

)

51

Figure 24 In the graph, the surface tension is shown as a function of temperature (℃) for all

concentrations. At, 10, 31.6, and 100 nM the surface tension did not decrease at the time

measured, while at 0.2, 0.316, 1.0, and 50 µM the surface tension had reached meso-

equilibrium.

20

30

40

50

60

70

80

90

20 30 40 50 60 70 80 90

Water

Experimentalwater 50 µM

1 µM

0.316

0.2 µM

0.1 µM

31.6 nM

10 nM

Temperature (℃)

Surf

ace

Ten

sio

n (

mN

/m)

A.

52

20

30

40

50

60

70

80

90

1 1.5 2 2.5 3 3.5

25 ℃ 0.2 uM

0.316 uM

1 uM

50 uM

Surf

ace

Ten

sio

n (

mN

/m)

log(time (s))

A.

20

30

40

50

60

70

80

90

1 1.5 2 2.5 3 3.5 4

30 ℃ 0.2 uM

0.316 uM

1 uM

50 uM

Surf

ace

Ten

sio

n (

mN

/m)

log(time (s))

B.

53

Figure 25 A) Measured surface tension as a function of log(time(s)) for 0.2, 0.316, 1, 50 μM solution of

ELP-foldon at pH of 7.4 and a salt concentration of 25 mM PBS at 25 ℃ . Experimental data is fit by

the Hua-Rosen equation to create theoretical curve ( ). B) Measured surface tension as a function of

time for 0.2, 0.316, 1, 50 μM solution of ELP-foldon at pH of 7.4 and a salt concentration of 25 mM

PBS at 30 ℃ C) Measured surface tension as a function of time for 0.2, 0.316, 1, 50 μM solution of

ELP-foldon at pH of 7.4 and a salt concentration of 25 mM PBS at 35 ℃.

20

30

40

50

60

70

80

90

1 1.5 2 2.5 3 3.5

35 ℃ 0.2 uM

0.316 uM

1 uM

50 uMSu

rfac

e T

ensi

on

(m

N/m

)

log(time (s))

C.

54

At different concentrations and constant temperature, the surface tension depends

on time (Figure 25). At 25, 30 and 35 ℃, and at 0.2, 0.316 and 1 μM concentrations,

formation of the region I, II, III are observed; however, at higher concentration (50 μM),

a direct transition to region III is observed. While concentration increases, lag time

decreases. When the temperature is increased, the lag time to reach meso-equilibrium

surface tension value decreases. As it is seen in the graphs, the meso-equilibrium surface

tension value does not change with the concentration or the temperature.

Figure 26 The half time (t*) as a function of concentration for 0.2, 0.316, 1, 50 μM solution of ELP-

foldon at pH of 7.4 and a salt concentration of 25 mM PBS at 25, 30, 35 ℃ . The slope is measured as -

1.2 for graph A and B, and for graph C, is -1.3.

y = -1.1474x + 2.2182

0

0.5

1

1.5

2

2.5

3

3.5

-1 0 1 2

log(

t*(s

))

log(Cb/uM)

25 ℃ A.

y = -1.1778x + 2.1465

0

0.5

1

1.5

2

2.5

3

3.5

-1 -0.5 0 0.5 1 1.5 2

log(

t*(s

))

log(Cb/uM)

30 ℃ B.

y = -1.3063x + 2.0275

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

-1 -0.5 0 0.5 1 1.5 2

log(

t* (s)

)

log(Cb/uM)

35 ℃ C.

55

Time dependent surface tension measurement of polymer solution can be fit to

find dynamic surface tension parameters (Table 3.2) including initial surface tension (𝛾𝑠)

when the drop is formed, meso-equilibrium surface tension (𝛾𝑚), the time the surface

tension 𝛾(𝑡) is half-way between 𝛾𝑠 and 𝛾𝑚, (𝑡∗) and empirical constant (n) using the

Hua-Rosen equation [13, 45]:

log𝛾𝑠 − 𝛾(𝑡)

𝛾(𝑡) − 𝛾𝑚= 𝑛 log

𝑡

𝑡∗

The t* value from the Hua-Rosen equation 𝑠cales with the bulk concentration with the

exponent m (Figure 26):

𝑡 ∝ 𝐶𝑏𝑚

According to Ward and Tordai, for diffusion controlled adsorption kinetics, the scaling

exponent is -2 [13]. In our study, the slope obtained for (ELP)40-foldon is -1.2 and -1.3 at

25 and 30 ℃ and 35 ℃ respectively (Figure 26). Since our experimental slope is not close

to -2, the adsorption kinetics is not exclusively diffusion controlled. The difference

between the the diffusion controlled slope and experimental slope can be explained by

the occurrence of polymer adsorption/desorption barriers [12].

The surface pressure at the meso-equilibrium is found to be ~23 mN/m. This

surface pressure does not depend on the temperature or the concentration.

56

Table 3.2 Parameters of the dynamic surface tension of (ELP)-Foldon.

3.1 ELP-Foldon’s Diffusion

Temperature

℃

Concentration

(μM)

𝒕∗

(s) 𝜸𝒔

(mN/m)

𝜸𝒎

(mN/m)

𝜸𝒔 − 𝜸𝒎

(mN/m)

n

25

0.2 927

71.9 ±1.5 49.7 ±1.5 22.3 4.8 0.316 676

1 177

50 1.8

30

0.2 1046

70.9 ±1.5 49.2 ±1.5 22 4.8 0.316 452

1 152

50 1.4

35

0.2 772

69.5 ±1.5 47.5 ±1.5 22.9 4.8 0.316 429

1 145

50 0.6

Figure 27 The figures show the movement of the polymer to the air-aqueous surface. a 0.316 μM

solution of ELP-foldon at pH 7.4 and a salt concentration of 25 mM PBS below the transition

temperature. A) The polymers amount is not enough to reduce surface tension hence the surface tension

value is approximately equal to PBS’s surface tension, region I. B) The surface has enough polymers to

reduce surface tension, region II. C) The surface has reached enough polymers’ amount to form

multilayer, region III.

AirA.

PBS 25 mM Salt Conc.

Surface Tension

AirB.

PBS 25 mM Salt Conc.

Surface Tension

PBS 25 mM Salt Conc.

Surface Tension

57

The polymer moves from one point to another point by diffusion through

Brownian motion in the aqueous solution. The polymers accumulate at the air-water

interface because the adsorbed protein relults in a lower energy system. The movement of

the polymer to the interface requires time. The time required is inversely proportional to

the diffusion velocity [46]. Figure 27 illustrates the regions (I, II, III) formation of the

(ELP)40-Foldon at a 0.316 μM solution . Region I is the induction time and at 25 °C is

between 0.5 and 5.5 minutes (0 minute is the solution’s first interaction time with air),

surface tension is closer to pure water surface tension and since there is not enough, the

polymers move to the air-aqueous surface, as illustrated in Figure 27A. Region II is rapid

fall region, at 25 °C, between 5.5 and 15.5 minutes, decreasing of the surface tension is

observed since enough polymers have been arrived to the surface forming a monolayer,

illustrated in Figure 27B. Region III is meso-equilibrium region, after 15.5 minutes, the

surface tension change is not observed since the surface reaches enough polymer

saturation forming a multilayer, illustrated in Figure 27C.

58

CHAPTER IV

CONCLUSIONS

Surface tension of the polymer ELP-Foldon in PBS solution was measured as a

function of temperature, time, and various concentration using a srop shape tensiometer.

At lower concentrations (10, 31.6 and 100 𝑛𝑀), the surface tension was approximately

equal to PBS surface tension as a function of temperature. Therefore, effect of the

polymer on the surface tension change can be thought negligible at the lower polymer

concentration. At higher concentrations (0.2, 0.316, 1 𝑎𝑛𝑑 50 𝜇𝑀), the surface tension

was reduced by the polymer. However, the decrease was observed to be dependent on

time, exhibiting three characteristic regions. It is noteworthy that at all the concentrations

and temperatures, the surface tension values were approximately equal to each other

around 49 ± 1.5 𝑚𝑁/𝑚 resulting in a surface pressure of ~23 mN/m that does not vary

with concentration or temperature. It is also observed that elapsed time to reach meso-

equilibrium surface tension was dependent upon the concentration and temperature.

When temperature and the concentration were increased, the time to reach equilibrium

decreased since polymer diffusion is increased by temperature, and the probability of the

presence of the polymer in the region close to the surface is increased by the

59

concentration; thus, they are adsorbed more rapidly since distance to the surface is

decreased.

The half time (t*) is shown to scale with concentration with an exponent of -1.2

and -1.3 at 25 and 30 ℃ and 35 ℃, respectively. This suggests that the ELP-Foldon

adsorption kinetics does not show exclusively diffusion controlled behavior.

The ELP-Foldon does not show c.m.c. formation point since the polymer-PBS

solution surface tension is not affected by the polymer concentration. The concentration

affects only the region formation. At higher concentration, regime formation is not

observed.

60

BIBLIOGRAPHY

[1] H. Alanazi, "Characterization Of Elastin-Like Polypeptides Using Viscometry,"

Cleveland State University, Cleveland, 2010.

[2] D. W. Urry, "Physical Chemistry of Biological Free Energy Transduction As

Demonstrated by Elastic Protein-Based Polymers," Journal of Physical Chemistry B,

pp. 11007-11028, 1997.

[3] D. L. Nettles, A. Chilkoti and L. A. Setton, "Applications of elastin-like

polypeptides in tissue engineering, Advanced Drug Delivery Reviews," Advanced

Drug Delivery Reviews 62, p. 1479–1485, 2010.

[4] J. J. Kelly, "Journal of Macromolecular Science," Part C—Polymer Reviews, vol.

C42, no. 3, p. 355–371, 2002.

[5] T. E. Creighton, Proteins Structure and Molecular Properties 2nd edition, New York:

W.H Freeman and Company , 1993.

[6] C. Beverung, C. Radke and H. Blanch, "Protein adsorption at the oil/water interface:

characterization of adsorption kinetics by dynamic interfacial tension

measurements," Beverung, C. Biophysical Chemistry, vol. 81, no. 1, pp. 59-80, 1999.

[7] B. Stückrad, W. J. Hiller and T. A. Kowalewski, "Measurement of dynamic surface

tension by the oscillating droplet method," Experiments in Fluids 15, pp. 332-340,

2003.

[8] A. Ciferri, "Charge-dependent and charge-independent contributions to ion-protein

interaction," Biopolymers, pp. 700 - 709, 2008.

[9] A. Ward and L. Tordai, "Time-dependence of boundary tensions of solutions. I. The

role of diffusion in time-effects," J. Chem. Phys., pp. 453-461, 1947.

[10] V. kazakov, O. Sinyachenko, V. Fainerman, U. Pison and R. Miller, Dynamic

Surface Tensiometry in Medicine, Amsterdam: Elsevier Science B.V., 2000.

[11] X. Y. Hua and M. J. Rosen, "Dynamic Surface Tension of Aqueous Surfactant

Solutions," Journal of Colloid and Interface Science, vol. 124, no. 2, p. 652–659,

1987.

61

[12] T. Gilányi, I. Varga, M. Gilányi and R. Mészáros, "Adsorption of poly(ethylene

oxide) at the air/water interface: A dynamic and static surface tension study,"

Journal of Colloid and Interface Science, vol. 301, pp. 428-435, 2006.

[13] V. P. Gilcreest, K. A. Dawson and A. V. Gorelov, "Adsorption Kinetics of NIPAM-

Based Polymers at the Air-Water Interface As Studied by Pendant Drop and Bubble

Tensiometry," J.Phys.Chem., vol. 110, no. 43, pp. 21903-21910, 2006.

[14] Ü. Geçgel, "Polimer-Surfaktant Etkileşimi," Trakya University , Edirne, 2008.

[15] Z. Sezgin, N. Yuksel and T. Baykara, "Preparation and Characterization Of

Polymeric Micelles As Drug Carrier System," J. Fac. Pharm, vol. 32, no. 2, pp. 125-

142, 2003.

[16] R. J. Hunter, Foundations Of Colloid Science Volume I., New York: Oxford

University Press, 1991.

[17] J. Z. Manojlovic , "The Krafft Temperature of Surfactant Solutions," Thermal

Science, vol. 16, no. 2, pp. S631-S640, 2012.

[18] J. P. Gavin, "Controlling the Size and Shape of Polypeptide Colloidal Particles:

Temperature Dependence of Particle Formation," Undergraduate Research Posters

2013. Book 17, 2013.

[19] A. Ghoorchian and N. B. Holland, "Molecular Architecture Influences the Thermally

Induced Aggregation Behavior of Elastin-like Polypeptides," Biomacromolecules,

vol. 12, no. 11, p. 4022−4029, 2011.

[20] A. Ghoorchian, K. Vandemark, K. Freeman, S. Kambow, N. B. Holland and K. A.

Streletzky, "Size and Shape Characterization of Thermoreversible Micelles of," The

Journal of Physical Chemistry B, vol. 117, p. 8865−8874, 2013.

[21] H. Iyota and R. Krastev, "Miscibility of sodium chloride and sodium dodecyl sulfate

in the adsorbed film and aggregate," Colloid and Polym Sci., vol. 287, no. 4, p. 425–

433, 2009.

[22] D. Myers, Surfactant Science and Technology, New York: VCH Publishers, Inc.,

1988.

[23] J. P. Gavin, M. G. Price and J. Mino, "Controlling micelle formation using mixtures

of linear and foldon-capped polypeptides (ELP): Measurements with UV-vis

spectroscopy," Undergraduate Research Posters 2014, Cleveland, 2014.

62

[24] S. Gill and P. H. Hippel, " Calculation of protein extinction coefficients from amino

acid sequence data," Anal. Biochem., vol. 182, pp. 319-26, 10 1989.

[25] A. R. Albis and A. F. Rincón, "Young-Laplace Equation In Convenient Polar

Coordinates And Its Implementation In Matlab, Revista Colombiana De Química,"

Rev.Colomb.Quim., vol. 39, no. 3, 2010.

[26] M. Hoorfar and A. W. Neumann, "Recent progress in Axisymmetric Drop Shape

Analysis (ADSA)," Advances in Colloid and Interface Science, vol. 121, p. 25–49,

2006.

[27] "ramé-hart Index of Accessories for Goniometer/Tensiometers Instruments," ramé-

hart instrument co, 2015. [Online]. Available:

http://www.ramehart.com/accessories.htm#100-22.

[28] F. Bashforth and J. C. Adams, An attempt to test the theory of capillary action,

Cambridge, 1892.

[29] d. R. O.I., On the Generalization of Axisymmetric Drop Shape Analysis, Toronto:

University of Toronto, 1993.

[30] S. Lahooti, O. del Río, P. Cheng and A. W. Neumann, Axisymmetric Drop Shape

Analysis (ADSA), New York: Marcel Dekker Inc, 1996.

[31] O. I. del Río and W. Neumann, "Axisymmetric Drop Shape Analysis:

Computational Methods for the Measurement of Interfacial Properties from the

Shape and Dimensions of Pendant and Sessile Drops," Journal Of Colloid And

Interface Science, vol. 196, p. 136–147, 1997.

[32] P. Cheng, Automation of Axisymmetric Drop Shape Analysis Using Digital Image

Processing, Toronto: University of Toronto, 1990.

[33] P. Cheng, D. Li, L. Boruvka, Y. Rotenberg and A. W. Neumann, "Colloids Surf,"

vol. 43, pp. 151-167, 1990.

[34] R. Maini and H. Aggarwal, "Study and Comparison of Various Image Edge

Detection Techniques," International Journal of Image Processing (IJIP), vol. 3, no.

1, 2009.

[35] T. Q. Pham, "Non-maximum Suppression Using fewer than Two Comparisons per

Pixel," Canon Information Systems Research Australia (CiSRA).

63

[36] A. Kalantarian, "Development Of Axisymmetric Drop Shape Analysis - No Apex

(ADSA-NA)," Doctor of Philosophy Graduate Department of Mechanical and

Industrial Engineering University of Toronto, Toronto, 2011.

[37] A. F. Staldera, T. Melchiorb, M. Müllerb, D. Saged, T. Bluc and M. Unserd, "Low-

bond axisymmetric drop shape analysis for surface tension and contact angle

measurements of sessile drops," Colloids and Surfaces A: Physicochem. Eng., p. 72–

81, 2010.

[38] J. C. Butcher, Numerical Methods for Ordinary Differential Equations 2nd edition,

New Zealand: Wiley, 2008.

[39] R. Bulirsch and J. Stoer, "Numerical treatment of ordinary differential equations by

extrapolation methods. Numerische Mathematik," Numerische Mathematik, vol. 8,

no. 1, pp. 1-13, 1966.

[40] S. Josef and R. Bulirsch, Introduction to Numerical Analysis, New York: Springer-

Verlag, 2002.

[41] J. Nelder and R. Mead, "A simplex method for function optimization," Computer

Journal, vol. 7, p. 308–313, 1965.

[42] Y. Rotenberg, L. Boruvka and A. W. Neumann, "Determination of surface tension

and contact angle from the shapes of axisymmetric fluid interfaces," Journal of

Colloid and Interface Science, vol. 93, no. 1, p. 169–183, 1983.

[43] J. H. Mathews and . F. K. Kurtis, "Numerical Optimization," in Numerical Methods

Using Matlab 4th Edition, Pearson, 2004, pp. 430- 436.

[44] N. B. Vargaftik, B. N. Volkov and L. D. Voljak, "International tables of the Surface

tension of water," J. Phys. Chem. Ref. Data, vol. 12, no. 3, 1983.

[45] J. Zhang and R. Pelton, "The dynamic behavior of poly(N-isopropylacrylamide) at

the air/water interface," Colloid and Surface A: Physicochemical and Engineering

Aspects, vol. 156, pp. 111-122, 1999.

[46] A. Ghoorchian, K. Vandemark, K. Freeman, S. Kambow, N. B. Holland and K. A.

Streletzky, "Size and Shape Characterization of Thermoreversible Micelles of Three-

Armed Star Elastin-Like Polypeptides," J. Phys. Chem. B, vol. 117, no. 29, p.

8865−8874, 2013.

[47] C. P., "Automation of Axisymmetric Drop Shape Analysis Using Digital Image

64

Processing," University of Toronto, Toronto, 1990.

[48] A. Ghoorchian, J. T. Cole and N. B. Holland, "Thermoreversible Micelle Formation

Using a Three-Armed Star Elastin-like Polypeptide," Macromolecules, vol. 43, no.

9, p. 4340–4345, 2010.

[49] L. H. Sperling, Introduction to Physical Polymer Science 4th Edition, New Jersey:

Wiley, 2006.

[50] D. E. Meyer and A. Chilkoti, "Genetically encoded synthesis of protein-based

polymers with precisely specified molecular weight and sequence by recursive

directional ligation: examples from the elastin-like polypeptide system,"

Biomacromolecules, vol. 3, no. 2, pp. 357-67, 2002 .

[51] SDS-Polyacrylamide Gel Electrophoresis (SDS-PAGE), Gainesville: EnCor

Biotechnology Inc., 2015.

[52] E. Y. Arashiro and N. R. Demarquette, "Use of the Pendant Drop Method to

Measure Interfacial Tension," Materials Research, vol. 2, no. 1, pp. 23-32, 1999.

65

APPENDIX

Experimental images were analyzed by the MatLab® codes to measure the

surface tension.

% pd_run.m % run file for pendant drop analysis

% c = (del_row*g)/surface tension

clc; clear;

% Input parameters for Pendent Drop Analysis

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%% Experimental Image Parameters %%%%%%%%%%%%%%%%%%%%%% % Image for Analysis (Color or Greyscale) img = 'IMAGE.jpg';

% edge detection parameters for edge detector % ed_type = 1 for c10anny % ed_type = 2 for sobel errors??????? % ed_type = 3 for bwboundaries with 'noholes' errors??? ed_type = 1; thresh = []; sigma = 1;

% number of rows to search for start of bwtraceboundary line_check = 10;

% calculate capillary diameter % cap_dia_points = # of points to search and average for capillary

diameter cap_dia_points = 20;

% rotate experimental data % rotation = 1 too use rotated experimental data rotation = 1;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%% Theoretical Parameters %%%%%%%%%%%%%% % Parameters c_start = -0.11347; % mm-2 b_start = 1.31989; % mm cap_dia_units = 0.72; % mm (1.27)(0.72)

del_row = 0.001; % g/mm3 g = -9806.4; % mm/s2

% choose optimization solver type % solver_type = 0 to exit without optimization % solver_type = 1 for fminsearch (Nelder-Mead) % solver_type = 2 for Levenberg-Marquardt

66

% maxiter is number of solver loops solver_type = 1; maxiter = 1;

% include residual plot % residual_plot = 0 for No % residual_plot = 1 for Yes residual_plot = 1;

% include error suface plot % err_sur_plot = 0 for No % err_sur_plot = 1 for Yes % deltab is the +- distance to vary b % deltac is the +- distance to vary c % nop is the number of data points for each delta err_sur = 0; deltab = 0.05; % careful changing this, may cause problems deltac = 0.05; % careful changing this, may cause problems nop = 5;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% define all parameters parameters(1) = ed_type; parameters(2) = sigma; parameters(3) = line_check; parameters(4) = solver_type; parameters(5) = maxiter; parameters(6) = del_row; parameters(7) = g; parameters(8) = cap_dia_points; parameters(9) = cap_dia_units; parameters(10) = residual_plot; parameters(11) = rotation;

% define error surface parameters err_surf_para(1) = err_sur; err_surf_para(2) = deltab; err_surf_para(3) = deltac; err_surf_para(4) = nop;

% directing function file colonal(img, c_start, b_start, parameters, thresh, err_surf_para);

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Volume_calc.m

function [vol_total] = vol_calc(x_theor_final, z_theor_final, b2)

x_theor_final_dim = x_theor_final.*b2; z_theor_final_dim = z_theor_final.*b2;

67

%[num] = xlswrite('xz_theor_final_dim.xls', [x_theor_final_dim

z_theor_final_dim]); % write to file

vol_total =

sum((pi.*((x_theor_final_dim(2:end,:).^2))).*(diff(z_theor_final_dim)))

;

return;

%%%unique_stable.m

function val = unique_stable(input_matrix)

% remove duplicate rows [values index] = unique(input_matrix, 'rows','first');

% resort data as 'stable' out = sortrows([index values]); val = out(:,2:3); return;

% unique_extract.m

function val = unique_extract(input_matrix)

% remove first bwtraceboundary data val1(1,:) = input_matrix(1,:); % adds first value to new matrix i = 2; while input_matrix(1,1) ~= input_matrix(i,1) val1(i,:) = input_matrix(i,:); i = i + 1; end

val1(i,:) = input_matrix(i,:); % adds last value to new matrix

%disp(val1)

% remove duplicate rows [values index] = unique(val1, 'rows','first');

% re-sort data as 'stable' out = sortrows([index values]); val = out(:,2:3);

return;

% sur_ten.m % function calculates surface tension from c, del_row, and g

function st = sur_ten(c_value, parameters3)

del_row = parameters3(1);

68

g = parameters3(2);

st = (del_row*g)/c_value;

return;

% residual.m % function to plot residual

function residual(c2, b2, x_new, z_new, ratio, height, residual_plot)

if residual_plot == 1 fprintf('\n Calculating Residual....\n'); % plot residual V(1) = c2; V(2) = b2; [dum, zx_residual] = obj_fun(V, x_new, z_new, ratio, height);

figure(11); plot(zx_residual(:,2), zx_residual(:,1)); xlabel('Drop Position'); ylabel('Residuals'); title('Residual Plot'); fprintf(' Residual Plot Complete....\n\n'); end

return;

function [mid_return, apex, height, cap_radius, theta, mid] =

pd_sym(b_matrix, top_cut_off, points, sigma) % this function finds the vertical symetric axis % of a pendent drop image curve % % input requires a matrix of [j i] or [y x] along with % the top cutoff value for scale and number of analysis % points

% output is of the matrix type [j i] or [y x]

% begin by finding approx vertical height j_max = max(b_matrix(:,1)); height = j_max - top_cut_off; % in pixels

% find midpoint of several horizontal values % construct loop statement to find midpoint values

i = 0; while (length(b_matrix(:,2))- i) > i mid(i+1,1) = (b_matrix(end-i,2) + b_matrix(i+1,2))/2; i = i + 1; end mid_coor = [mid b_matrix(1:length(mid),1)]; % [x y]

69

% check for outliers then eliminate s = std(mid_coor(:,1)); min_x = mean(mid_coor(:,1)) - sigma*s; max_x = mean(mid_coor(:,1)) + sigma*s;

sss = mid_coor(:,1) >= min_x & mid_coor(:,1) <= max_x; mid_sorted = mid_coor(sss,:);

a1 = length(mid_coor(:,1)); a2 = length(mid_sorted(:,1));

% fit a line to remaining data point p = polyfit(mid_sorted(:,1), mid_sorted(:,2), 1);

% calc fitted values midX = linspace(min(mid_coor(:,1))-1, max(mid_coor(:,1))+1, points); midY = polyval(p, midX);

% format return mid_return = [midY; midX]';

% calc vertex position intersect = (j_max - p(2))/p(1); apex = [intersect j_max];

% calc cap_radius position cap = (top_cut_off - p(2))/p(1); cap_apex = [cap top_cut_off]; cap_radius = b_matrix(end,2) - cap; cap_radius2 = (b_matrix(end,2) - b_matrix(1,2))/2;

% calc theta theta = (atan(1/(-1*p(1))));

% output results to screen fprintf(' Results from Symmetric Check\n'); fprintf('Standard Deviation = %.4f\n', s); fprintf('Number of mid-points before sort = %.0f\n', a1); fprintf('Number of mid-points after sort = %.0f\n', a2); fprintf('Offset Angle = %5.4f degrees\n', (theta*(180/pi))); fprintf('Slope = %5.4f\n', (-1*p(1))); fprintf('Intercept = %5.4f\n', p(2)); fprintf('Apex at position [x z] = %5.4f %5.4f\n', apex(1), apex(2)); fprintf('Cap_Center at position [x z] = %5.4f %5.4f\n', cap_apex(1),

cap_apex(2)); fprintf('Cap_Radius = %5.4f pixels (uses line equ)\n', cap_radius); fprintf('Cap_Radius = %5.4f pixels\n', cap_radius2); fprintf('\nPress any Key to Accept Results and Continue or CTRL-C to

exit....\n');

return;

% optim_solver.m % includes functions to optimize c and b parameters for curve fitting % first method is that of Nelder-Mead

70

% second method is Levenberg-Marquardt solver % can change TolX and TolFun as required

function [c2, b2] = optim_solver(x_new, z_new, parameters2)

% define parameters solver_type = parameters2(1); c_start = parameters2(2); b_start = parameters2(3); maxiter = parameters2(4); ratio = parameters2(5); height = parameters2(6); del_row = parameters2(7); g = parameters2(8);

switch (solver_type) case (0) c2 = c_start; b2 = b_start; return;

case (1) % % minimize objective function to find solution % % optimization using fminsearch fprintf('Press Enter to Continue with Fitting (Nelder-Mead)\n'); fprintf('Initial c = %5.4f\n', c_start); fprintf('Initial b = %5.4f\n', b_start); in_flag = input(' ');

results = zeros(maxiter,5); options = optimset('Display','iter','TolX',1e-4,'TolFun',1e-4); tic; % start timer for iter = 1:maxiter

i_guess = [c_start b_start]; [c_b, fval] = fminsearch(@(V) obj_fun(V, x_new, z_new, ratio,

height), i_guess, options);

fprintf('\nFinal c = %5.10f\n', c_b(1)); fprintf('Final b = %5.10f\n', c_b(2)); fprintf('Error Function Value = %5.10f\n', fval);

results(iter,1) = iter; % iteration results(iter,2) = c_b(1); % c value results(iter,3) = c_b(2); % b value results(iter,4) = fval; % error

% calculate surface tension parameters3(1) = del_row; parameters3(2) = g; results(iter,5) = sur_ten(c_b(1), parameters3);

% reload c_start = c_b(1); b_start = c_b(2);

71

end toc; % stop timer

case(2) % Levenberg-Marquardt solver fprintf('\nContinue with Fitting (Levenberg-Marquardt)\n'); fprintf('Initial c = %5.4f\n', c_start); fprintf('Initial b = %5.4f\n', b_start); in_flag = input(' ');

results = zeros(maxiter,5); options1 = optimset('Algorithm','levenberg-

marquardt','ScaleProblem','Jacobian','Display','iter','TolX',1e-

4,'TolFun',1e-4); tic; % start timer for iter = 1:maxiter

i_guess = [c_start b_start]; [c_b, resnorm] = lsqnonlin(@(V) obj_funb(V, x_new, z_new,

ratio, height), i_guess, [], [], options1);

fprintf('\nFinal c = %5.10f\n', c_b(1)); fprintf('Final b = %5.10f\n', c_b(2)); fprintf('Error Function Value = %5.10f\n', resnorm);

results(iter,1) = iter; results(iter,2) = c_b(1); results(iter,3) = c_b(2); results(iter,4) = resnorm;

% calculate surface tension parameters3(1) = del_row; parameters3(2) = g; results(iter,5) = sur_ten(c_b(1), parameters3);

% reload c_start = c_b(1); b_start = c_b(2);

end toc; % stop timer end

% display results fprintf('\nresults\n'); fprintf(' Iteration c (mm-2) b (mm) Error Surface Tension

(mN/m)\n'); disp(results);

% show plot of both curves c2 = c_b(1); b2 = c_b(2);

% generate plot of final results [x_new_final, z_new_final] = dim2dimless(x_new, z_new, ratio, b2);

72

[x_theor_final, z_theor_final] = lap_run(c2, b2, height, ratio);

figure(9); plot(x_theor_final, z_theor_final,'r.','MarkerSize', 0.1); xlabel('x Dimensionless'); ylabel('z Dimensionless'); title('Final Results'); axis equal; hold on plot(x_new_final, z_new_final,'k.','MarkerSize', 0.1); legend('Theoretical','Experimental'); hold off

% calc volume vol = vol_calc(x_theor_final, z_theor_final, b2); fprintf('\nVolume = %5.4f mm3\n', vol);

return;

% obj_funb.m % original objective function % used for Levenberg-Marquardt method

function obj = obj_funb(V, x_new, z_new, ratio, height)

c = V(1); b = V(2);

% convert to dimensionless [x_new_dl, z_new_dl] = dim2dimless(x_new, z_new, ratio, b);

exp_data(:,1) = z_new_dl; exp_data(:,2) = x_new_dl;

% call function lap_run.m [x_theor, z_theor] = lap_run(c, b, height, ratio);

theor_data(:,1) = z_theor; theor_data(:,2) = x_theor;

% calculate shortest distance results = zeros(length(exp_data(:,1)),4);

for j = 1:length(exp_data(:,1)) term1 = exp_data(j,1) - theor_data(:,1); % z term2 = exp_data(j,2) - theor_data(:,2); % x dist = sqrt((term1.^2)+(term2.^2));

[dum, idx] = min(dist); % min distance index value

results(j,1) = exp_data(j,1); results(j,2) = exp_data(j,2); results(j,3) = theor_data(idx,1); results(j,4) = theor_data(idx,2);

73

end

distzx(:,1) = (results(:,1) - results(:,3)); distzx(:,2) = (results(:,2) - results(:,4));

obj = [distzx(:,1); distzx(:,2)]; return;

% obj_fun.m

%original objective function % used for Nelder-Mead method

function [obj, zx_residual] = obj_fun(V, x_new, z_new, ratio, height)

c = V(1); b = V(2);

%convert to dimensionless [x z] [exp_data(:,2), exp_data(:,1)] = dim2dimless(x_new, z_new, ratio, b);

% call function lap_run.m [x_theor, z_theor] = lap_run(c, b, height, ratio);

theor_data(:,1) = z_theor; theor_data(:,2) = x_theor;

mindist = zeros(length(exp_data(:,1)),1); zx_residual = zeros(length(exp_data(:,1)),2);

for j = 1:length(exp_data(:,1)) term1 = exp_data(j,1) - theor_data(:,1); % z term2 = exp_data(j,2) - theor_data(:,2); % x dist = sqrt((term1.^2)+(term2.^2)); [mindist(j,1), idx] = min(dist); % min distance index value

terma = exp_data(j,1) - theor_data(idx,1); termb = exp_data(j,2) - theor_data(idx,2);

if terma > 0 && termb < 0 zx_residual(j,1) = mindist(j,1)*-1; else zx_residual(j,1) = mindist(j,1); end zx_residual(j,2) = j;

end

obj = sum((mindist(:,1)).^2);

return;

74

% lap_run.m

function [x2 z2] = lap_run(c, b, height, ratio)

para(1) = c; para(2) = b;

% conditions for the ODE solver as [x z phi] s_span = [0:0.001:8]; y_initial = [1e-20, 0, 0];

% call ODE solver as [x z phi] [S, Y] = ode45(@lap_equ, s_span, y_initial, [], para);

% sort data i = 1; while Y(i,2) < Y(i+1,2) && Y(i,1) < Y(i+1,1); i = i + 1; end

while Y(i,2) < Y(i+1,2) && Y(i,1) >= Y(i+1,1); i = i + 1; end

% Sort Theor. Data x(:,1) = Y(1:i,1); z(:,1) = Y(1:i,2); s(:,1) = S(1:i,1); Phi(:,1) = Y(1:i,3)*(180/pi);

% match height of experimetal curve height_dl = height*ratio/b; % convert to dimensionless height z_index = z(:,1) <= height_dl; x2(:,1) = x(z_index,1); z2(:,1) = z(z_index,1);

return;

% lap_equ.m % function file for pendant drop analysis (lap_run.m)

function ydot = lap_equ(s, f, para) c = para(1); b = para(2);

x = f(1); z = f(2); phi = f(3);

ydot(1) = (cos(phi)); ydot(2) = (sin(phi)); ydot(3) = 2 + c*z*(b^2) - ((sin(phi))/x);

75

ydot = ydot'; return;

% image_crop.m

function [d, top_cut_off] = image_crop(BW3, img)

input1 = 0; while input1 ~= 1

if input1 == 2 % clear all data clear top_cut_off d; clear figure 5; end

% construct menu fprintf('\n Enter Z-axes Height Cut-off\n'); fprintf('Press 1 for Known Input in Pixels\n'); fprintf('Press 2 for Graphical Analysis\n'); input2 = input(' ');

if input2 == 1 top_cut_off = input('\nEnter Pixel value for drop height: '); else % insert graphical analysis imtool(img); top_cut_off = input('\nEnter Pixel value for drop height: '); imtool close all; end

d = BW3(BW3(:,1)>=top_cut_off,:); % removes values above top_cut_off

% verify results figure(5); imshow(img); hold on; plot(d(:,2),d(:,1),'r.','MarkerSize', 0.1); hold off;

% verify correct info from user fprintf('\n Verify Image Cut off is Correct\n'); fprintf('Press 1 to Continue\n'); fprintf('Press 2 to Try Again\n'); fprintf('Press CTRL-C to Exit\n'); input1 = input(' ');

end imtool close all;

return;

76

%image_construct.m

function BW3 = image_construct(img, thresh, sigma, line_check, ed)

%load the image rgb_img = imread(img); figure(1); imshow(rgb_img); title('Original Image');

% Change image to grayscale then binary using threshhold level = graythresh(rgb_img); I2 = im2bw(rgb_img); figure(2); imshow(I2); % can also look at imtool title('Binary Image');

% detect all edges using MATLAB edge detector switch (ed) case (1) BW = edge(I2,'canny'); case (2) BW = edge(I2,'sobel'); case (3) BW = bwboundaries(I2,'noholes'); end

figure(3); imshow(BW); title('Image After Edge Detection');

% find the start of target profile edge for bwtraceboundary script s=size(BW); for row = 2:line_check % be careful with this, first row must only for col=1:s(2) % contain the start of the curve if BW(row,col) == 1 %disp([BW(row,col) row col]); e_switch = 1; break; end end if e_switch == 1 break; end

end %disp([row,col]);

input1 = 0; while input1 ~= 1

if input1 == 2 % clear all data and open BW display with input clear BW3 BW2; close figure 4;

77

% need to obtain [row col] for bwtrace fprintf('\n Use imtool to Select Start of Trace Function\n'); imtool(BW); col = input('Input col of Starting Point: '); row = input('\nInput row of Starting Point: '); imtool close all; end

% trace profile BW2 = bwtraceboundary(BW, [row,col], 'S'); %disp(BW2);

% remove duplicate rows BW3 = unique_extract(BW2); %BW3 = unique_stable(BW2);

% plot results for review figure(4) plot(BW3(:,2),BW3(:,1),'k.','MarkerSize', 0.1); axis ij; axis equal; title('Image After Boundary Trace');

% verify correct info from user fprintf('\n Verify Boundary Trace is Correct\n'); fprintf('Press 1 to Continue\n'); fprintf('Press 2 for Manual Control\n'); fprintf('Press CTRL-C to Exit\n'); input1 = input(' '); end

return;

% err_sur_plot.m % calculates and plots the error surface

function err_sur_plot(c2, b2, x_new, z_new, err_surf_para)

% define err_surf_para err_sur = err_surf_para(1); deltab = err_surf_para(2); deltac = err_surf_para(3); nop = err_surf_para(4); ratio = err_surf_para(5); height = err_surf_para(6);

if err_sur == 1 fprintf(' Constructing Error Surface....\n'); count = 0; countout = 0; xzerr = zeros(nop^2,3);

for b_span = linspace(b2-deltab, b2+deltab, nop); for c_span = linspace(c2-deltac, c2+deltac, nop);

78

V(1) = c_span; V(2) = b_span;

% uses Nelder-Mead obj function [obj_err, dum] = obj_fun(V, x_new, z_new, ratio, height);

% store data count = count + 1; xzerr(count,1) = V(2); xzerr(count,2) = V(1); xzerr(count,3) = log10(obj_err); end countout = countout + 1; fprintf(' %i', countout); end

% reshape data for plotting x = reshape(xzerr(:,1),nop,nop)'; y = reshape(xzerr(:,2),nop,nop)'; z = reshape(xzerr(:,3),nop,nop)';

fprintf(' Surface Complete....\n');

%plot xzerr surface figure(10); surfc(x, y, z); xlabel('b'); ylabel('c'); zlabel('Log Error'); title('Error Surface Plot'); end

return;

% edge_reposition.m

function [x_new, z_new] = edge_reposs(d, apex)

% reposition drop where apex = [0 0] and extract half for fitting % extract half the curve for comparison to theory

% idx = (d(:,2)>round(apex(1))); % obtains logical index values % d_half = d(idx,:); % % % reposition at apex = [0 0] % x_new = d_half(:,2) - apex(1); % use for evaluatiuon % z_new = (d_half(:,1) - apex(2)).*-1; % use for evaluation

%%%%%%%%%%%%%%% %try this

zx_zeroed(:,2) = d(:,2) - apex(1); % reposition x

79

zx_zeroed(:,1) = (d(:,1) - apex(2)).*-1; % reposition z

idx_rt = (zx_zeroed(:,2)>=0); idx_lt = (zx_zeroed(:,2)<=0); zx_new_rt = zx_zeroed(idx_rt,:); zx_new_lt = zx_zeroed(idx_lt,:);

x_new = zx_new_rt(:,2); z_new = zx_new_rt(:,1);

return;

% dim2dimless.m % converts image pixels into dimensionless units

function [x_new_dl, z_new_dl] = dim2dimless(x_new, z_new, ratio, b)

% convert to dimensionless z_new_dl = z_new.*ratio./b; x_new_dl = x_new.*ratio./b;

return;

% colonal.m % this function serves to call function for pendent drop shade analysis % also does some printing and screen output functions % input from pd_run.m

function colonal(img, c_start, b_start, parameters, thresh,

err_surf_para)

ed_type = parameters(1); sigma = parameters(2); line_check = parameters(3); solver_type = parameters(4); maxiter = parameters(5); del_row = parameters(6); g = parameters(7); cap_dia_points = parameters(8); cap_dia_units = parameters(9); residual_plot = parameters(10); rotation = parameters(11);

% image filtering and edge detection analysis % produce fig 1 - 4 fprintf('\n Beginning Edge Detection....\n'); BW3 = image_construct(img, thresh, sigma, line_check, ed_type); fprintf(' Edge Detection Complete....\n');

% calculate capillary diameter fprintf('\n Calculating Capillary Diameter....\n');

80

ratio = cap_dia_calc(BW3, cap_dia_points, cap_dia_units, img); fprintf(' Calculation Complete....\n');

% image crop selection of drop height (top_cut_off) % produce fig 5 fprintf('\n Beginning Edge Crop....\n'); [d, top_cut_off] = image_crop(BW3, img); fprintf(' Edge Crop Complete....\n');

% find vertical symetric axis call function pd_sym fprintf('\n Beginning Symmetric Check....\n\n'); points = 200; % number of analysis points to determine sym [mid_zx, apex, height, cap_radius, theta, mid] = pd_sym(d, top_cut_off,

points, sigma); fprintf(' Symmetric Check Complete....\n');

fprintf('\n Performing Edge Extraction and Axes Centering....\n'); [x_new, z_new] = edge_reposs(d, apex); %[num] = xlswrite('xz_new.xls', [x_new z_new]); % write to file

% rotate experimental data (pixels) R = [cos(theta) -sin(theta); sin(theta) cos(theta)]; xz_new_rot = (R*[x_new'; z_new'])';

% calc new height from rotation (pixels) height = max(xz_new_rot(:,2));

% convert original experimental data (pixels) to dimensionless form for

plotting [x_new_dl, z_new_dl] = dim2dimless(x_new, z_new, ratio, b_start); cap_radius_dl = cap_radius*ratio/b_start; fprintf(' Repositioning Complete....\n');

% convert rotated experimental data (pixels) to dimensionless form for

plotting [x_new_dl_rot, z_new_dl_rot] = dim2dimless(xz_new_rot(:,1),

xz_new_rot(:,2), ratio, b_start);

figure(6) imshow(img); hold on plot(d(:,2),d(:,1),'r.','MarkerSize', 0.1); plot(mid_zx(:,2),mid_zx(:,1),'b'); plot(mid, d(1:length(mid)),'-

r.','MarkerFaceColor','g','MarkerSize',0.2); hold off title('Drop Profile with Midpoint Line Detected');

figure(7) plot(x_new_dl, z_new_dl,'k.','MarkerSize', 0.1); xlabel('x Dimensionless'); ylabel('z Dimensionless'); axis equal; hold on plot(x_new_dl_rot, z_new_dl_rot,'g.','MarkerSize', 0.1);

81

title('Drop Profile Analysis Image Used for Fitting'); legend('Original','Rotated'); hold off

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%% Theor. Profile %%%%%%%%%%%%%%%%%%%%%%

% apex coordinates sent from image apex coordinates sent as [x z] [x_theor, z_theor] = lap_run(c_start, b_start, height, ratio);

figure(8) plot(x_theor, z_theor,'r.','MarkerSize', 0.1); xlabel('x Dimensionless'); ylabel('z Dimensionless'); axis equal; hold on plot(x_new_dl, z_new_dl,'k.','MarkerSize', 0.1); plot(x_new_dl_rot, z_new_dl_rot,'g.','MarkerSize', 0.1); hold off title('Initial Profiles'); legend('Theor','Experimental','Rotated');

% optimization methods % define parameters parameters2(1) = solver_type; parameters2(2) = c_start; parameters2(3) = b_start; parameters2(4) = maxiter; parameters2(5) = ratio; parameters2(6) = height; parameters2(7) = del_row; parameters2(8) = g;

% Use rotated Image? if rotation == 1 fprintf('\n Using Rotated Experimental Data\n'); x_new = xz_new_rot(:,1); z_new = xz_new_rot(:,2); end

%[c2, b2] = optim_solver(xz_new_rot(:,1), xz_new_rot(:,2),

parameters2); % rot data

% calculate error surface plot % plots fig 10 % call function for error plotting % err_sur_plot(c2, b2, xz_new_rot(:,1), xz_new_rot(:,2),

err_surf_para); % % residual plot % residual(c2, b2, x_new, z_new, ratio, height, residual_plot)

[c2, b2] = optim_solver(x_new, z_new, parameters2);

% calculate error surface plot % plots fig 10

82

% define err_surf_para err_surf_para(5) = ratio; err_surf_para(6) = height;

% call function for error plotting err_sur_plot(c2, b2, x_new, z_new, err_surf_para); % residual plot residual(c2, b2, x_new, z_new, ratio, height, residual_plot)

return;

% cap_dia_calc % calculates the capillary diameter for a selected % number of rows then determines the average

function ratio = cap_dia_calc(BW3, cap_dia_points, cap_dia_units, img)

% cap_dia = zeros(cap_dia_points,1); % for i = 1:cap_dia_points % idxBW3 = ismember(BW3(:,1),BW3(i,1)); % cap_dia(i,1) = diff(BW3(idxBW3,2)); % end % % % calculate average of all points % cap_dia_avg = mean(cap_dia);

%%%%%%%%%%%%%%%%%%%% % try this way

right = BW3(1:cap_dia_points,:); left = BW3(end-cap_dia_points+1:end,:);

sorted_data = intersect(right(:,1), left(:,1));

cap_dia = zeros(2,length(sorted_data(:,1))); for i = 1:length(sorted_data)

idxright = ismember(right(:,1),sorted_data(i)); avgright = mean(right(idxright,2)); cap_dia(1,i) = avgright;

idxleft = ismember(left(:,1),sorted_data(i)); avgleft = mean(left(idxleft,2)); cap_dia(2,i) = avgleft;

end

83

% calculate average of all points cap_dia_avg = mean(diff(cap_dia));

%end %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% calculate ratio ratio = cap_dia_units/cap_dia_avg;

% output results to screen fprintf(' Results from Capillary Diameter Calculation\n'); fprintf('Number of Points Selected = %5.0f points\n', cap_dia_points); fprintf('Average Capillary Diameter = %5.2f pixels\n', cap_dia_avg); fprintf('Calculated Ratio = %5.5f mm/pixel\n', ratio); disp(' '); fprintf(' Use the Calculated Capillary Diameter?\n'); input2 = 0; input2 = input('Press 1 for Yes or Press 2 for No: ');

if input2 == 2 imtool(img); cap_dia_avg = input('\nEnter Average Capillary Diameter in Pixels:

'); imtool close all; clear ratio; ratio = cap_dia_units/cap_dia_avg; fprintf('Calculated Ratio = %5.5f mm/pixel\n',ratio); end

return;